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MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.
The histogram shown gives an idea of
the shape of a normal distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The normal distribution describes many real-life data sets.
Although the normal distribution is a continuous distribution
whose graph is a smooth curve, an appropriate histogram can
give a very good approximation to the actual normal graph.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
5. The total area under the curve is 1.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
5. The total area under the curve is 1.
6. Roughly 68% of the data values
are within 1 standard deviation
of the mean, 95% are within 2
standard deviations of the mean
and 99.7% are within 3 standard
deviations of the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
5. The total area under the curve is 1.
6. Roughly 68% of the data values
are within 1 standard deviation
of the mean, 95% are within 2
standard deviations of the mean
and 99.7% are within 3 standard
deviations of the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
5. The total area under the curve is 1.
6. Roughly 68% of the data values
are within 1 standard deviation
of the mean, 95% are within 2
standard deviations of the mean
and 99.7% are within 3 standard
deviations of the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
Properties of the Normal Distribution
𝜇 is the mean and 𝜎 is the standard deviation of the distribution.
1. A normal curve is bell-shaped.
2. The highest point on the curve is at the mean.
3. The mean, median and mode are equal.
4. The curve is symmetric with respect to its mean.
5. The total area under the curve is 1.
6. Roughly 68% of the data values
are within 1 standard deviation
of the mean, 95% are within 2
standard deviations of the mean
and 99.7% are within 3 standard
deviations of the mean.
The 68-95-99.7 Rule for Normal Distributions
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
425
475
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
425
475
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
425
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
450 – 425 = 25
25
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
475 – 450 = 25
25 25
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
25
425
25
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
But 25 is the standard deviation.
25 25
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95% are within 2 standard deviations of the mean and
99.7% are with 3 standard deviations of the mean.
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
But 25 is the standard deviation.
25 25
425
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95%68-95-99.7
are within 2Rule
standard
of the
mean
and
The
tellsdeviations
us that 68%
of the
scores
99.7%
are with
3 standard
deviations
themean.
mean.
are within
1 standard
deviation
ofofthe
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
25
425
25
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95%68-95-99.7
are within 2Rule
standard
of the
mean
and
The
tellsdeviations
us that 68%
of the
scores
99.7%
are with
3 standard
deviations
themean.
mean.
are within
1 standard
deviation
ofofthe
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
68%
25
425
25
450
475
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95%68-95-99.7
are within 2Rule
standard
of the
mean
and
The
tellsdeviations
us that 68%
of the
scores
99.7%
are with
3 standard
deviations
themean.
mean.
are within
1 standard
deviation
ofofthe
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
68%
25
25
If there are 1000 scores, we would expect about
425
475
68% of them to be between 425 and 475.
450
The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95%68-95-99.7
are within 2Rule
standard
of the
mean
and
The
tellsdeviations
us that 68%
of the
scores
99.7%
are with
3 standard
deviations
themean.
mean.
are within
1 standard
deviation
ofofthe
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
68%
25
25
If there are 1000 scores, we would expect about
425
475
68% of them to be between 425 and 475.
450
1000 68% = 1000 0.68 = 680 The shaded area gives the probability of
a score falling in the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall between 425 and 475?
The 68-95-99.7 Rule
68% of the values are within 1 std. deviation of the mean,
95%68-95-99.7
are within 2Rule
standard
of the
mean
and
The
tellsdeviations
us that 68%
of the
scores
99.7%
are with
3 standard
deviations
themean.
mean.
are within
1 standard
deviation
ofofthe
Here 𝜇 = 𝟒𝟓𝟎(mean) and 𝜎 = 𝟐𝟓(std deviation)
The interval shown consists of all scores
within 1 standard deviation of the mean.
68%
25
25
If there are 1000 scores, we would expect about
425
475
68% of them to be between 425 and 475.
450
1000 68% = 1000 0.68 = 680 So, we expect about 680 scores to be in
the 425 – 475 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
450
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
Only the question has changed.
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 500 is from the mean.
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 500 is from the mean.
500 – 450 = 50
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 500 is from the mean.
But 25 is the standard deviation.
500 – 450 = 50
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 500 is from the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
But 25 is the standard deviation.
500 – 450 = 50
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This is exactly the
same
distribution
as before.
The
68-95-99.7
Rule
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
within
2 standard
of theofmean.
68%are
of the
values
are
withindeviations
1has
std. deviation
the mean,
Only
the
question
changed.
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
within
2 standard
of theofmean.
68%are
of the
values
are
withindeviations
1has
std. deviation
the mean,
Only
the
question
changed.
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
95%
50
400
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
That leaves
are
within
2
standard
deviations
of
the
mean.
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
5% to be split
95%we
areneed
withinto
2 standard
of the mean and
between the
First
find howdeviations
many standard
two tails.
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
95%
50
400
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
That leaves
are
within
2
standard
deviations
of
the
mean.
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
5% to be split
95%we
areneed
withinto
2 standard
of the mean and
between the
First
find howdeviations
many standard
two tails.
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
Half of 5% is 2.5%.
So the orange shaded area is 2.5%.
95%
50
400
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
That leaves
are
within
2
standard
deviations
of
the
mean.
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
5% to be split
95%we
areneed
withinto
2 standard
of the mean and
between the
First
find howdeviations
many standard
two tails.
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
95%
2.5%
Half of 5% is 2.5%.
So the orange shaded area is 2.5%.
50
400
2.5%
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
That leaves
are
within
2
standard
deviations
of
the
mean.
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
5% to be split
95%we
areneed
withinto
2 standard
of the mean and
between the
First
find howdeviations
many standard
two tails.
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
95%
2.5%
Half of 5% is 2.5%.
So the orange shaded area is 2.5%.
1000 2.5% = 1000 0.025 =25
50
400
2.5%
50
450
500
The shaded area gives the probability of
a score falling above 500.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall above 500?
This
exactly the
same
asthe
before.
Theis68-95-99.7
Rule
tellsdistribution
us that
95% of
scores
The
68-95-99.7
Rule
That leaves
are
within
2
standard
deviations
of
the
mean.
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
5% to be split
95%we
areneed
withinto
2 standard
of the mean and
between the
First
find howdeviations
many standard
two tails.
99.7% are500
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 500 is 2 standard deviations
(25 + 25 = 50) from the mean.
95%
2.5%
Half of 5% is 2.5%.
So the orange shaded area is 2.5%.
1000 2.5% = 1000 0.025 =25
50
400
2.5%
50
450
500
So,shaded
we expect
25 scores
to beof
The
areaabout
gives the
probability
a score fallingabove
in the500.
450 – 500 range.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 1 standard
deviation from the mean
68% of scores
𝜇−𝜎
𝜇
𝜇+𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 1 standard
deviation from the mean
68% of scores
34% 34%
𝜇−𝜎
𝜇
𝜇+𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 1 standard
deviation from the mean
16%
68% of scores
34% 34%
𝜇−𝜎
𝜇
16%
𝜇+𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 2 standard
deviations from the mean
95% of scores
𝜇 − 2𝜎
𝜇
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 2 standard
deviations from the mean
95% of scores
47.5%
47.5%
𝜇 − 2𝜎
𝜇
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 2 standard
deviations from the mean
2.5%
95% of scores
47.5%
47.5%
𝜇 − 2𝜎
𝜇
2.5%
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 3 standard
deviations from the mean
99.7% of scores
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 3 standard
deviations from the mean
99.7% of scores
49.85% 49.85%
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
This procedure can be somewhat simplified by noticing that the number
of standard deviations of the interval endpoints from the mean
determines the percentages under the normal curve.
When endpoints are 3 standard
deviations from the mean
0.15%
𝜇 − 3𝜎
99.7% of scores
49.85% 49.85%
𝜇
0.15%
𝜇 + 3𝜎
When endpoints are 1 standard
deviation from the mean
SUMMARY
How each
percentage
of the 68-95-99.7
rule breaks down
underneath
the Normal Curve
16%
68% of scores
34% 34%
16%
𝜇−𝜎
𝜇
𝜇+𝜎
When endpoints are 2 standard
deviations from the mean
2.5%
95% of scores
47.5%
47.5%
2.5%
𝜇 − 2𝜎
𝜇
𝜇 + 2𝜎
When endpoints are 3 standard
deviations from the mean
0.15%
𝜇 − 3𝜎
99.7% of scores
49.85% 49.85%
𝜇
0.15%
𝜇 + 3𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
450
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
Only the question has changed.
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 375 is from the mean.
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 375 is from the mean.
450 – 375 = 75
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 375 is from the mean.
But 25 is the standard deviation.
450 – 375 = 75
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the same distribution as before.
Only the question has changed.
First we need to find how many standard
deviations 375 is from the mean.
So 375 is 3 standard deviations
(25 + 25 + 25 =75) from the mean.
But 25 is the standard deviation.
450 – 375 = 75
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the
same
distribution
as before.
The
68-95-99.7
Rule
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
99.7% are375
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 375 is 3 standard deviations
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This is exactly the
same
distribution
as before.
The
68-95-99.7
Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
are
within
3
standard
deviations
of
the
mean.
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
99.7% are375
withis3from
standard
deviations
thedeviations
mean. of the mean.
So 375 is 3 standard deviations
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
The 68-95-99.7 Rule
let’s take
The
68-95-99.7
Rule
tells
us
that
99.7%
of
the
scores
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
thedeviations
mean. of the mean.
developed.
So 375 is 3 standard deviations
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
The 68-95-99.7 Rule
let’s take
The
68-95-99.7
Rule
tells
us
that
99.7%
of
the
scores
68% of the
values
within 1has
std. deviation
Only
theare
question
changed.of the mean,
advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean and
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
thedeviations
mean. of the mean.
developed.
So 375 is 3 standard deviations
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
When endpoints are 1 standard
deviation from the mean
SUMMARY
How each
percentage
of the 68-95-99.7
rule breaks down
underneath
the Normal Curve
16%
68% of scores
34% 34%
16%
𝜇−𝜎
𝜇
𝜇+𝜎
When endpoints are 2 standard
deviations from the mean
2.5%
95% of scores
47.5%
47.5%
2.5%
𝜇 − 2𝜎
𝜇
𝜇 + 2𝜎
When endpoints are 3 standard
deviations from the mean
0.15%
𝜇 − 3𝜎
99.7% of scores
49.85% 49.85%
𝜇
0.15%
𝜇 + 3𝜎
When endpoints are 1 standard
deviation from the mean
SUMMARY
How each
percentage
of the 68-95-99.7
rule breaks down
underneath
the Normal Curve
16%
68% of scores
34% 34%
16%
𝜇−𝜎
𝜇
𝜇+𝜎
When
endpoints are 2 standard
We figured out that 375 is
deviations
from
the mean
3 standard
deviations
of scores
from 95%
the mean,
so the
graph is 47.5%
the one 2.5%
2.5% bottom
47.5%
𝜇 − 2𝜎 that we𝜇need.
𝜇 + 2𝜎
When endpoints are 3 standard
deviations from the mean
0.15%
𝜇 − 3𝜎
99.7% of scores
49.85% 49.85%
𝜇
0.15%
𝜇 + 3𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
So the orange shaded area is 0.15%.
75
375
450
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
So the orange shaded area is 0.15%.
75
450
1000 0.15% = 1000 0.0015 =1.5 375
The shaded area gives the probability of
a score falling below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 1000 students on an intelligence test is a
normal distribution with mean 450 and standard deviation of 25.
How many scores do we expect to fall below 375?
This time,
This is exactly the
same
distribution
as
before.
When endpoints are 3 standardlet’s take
The 68-95-99.7 Rule
The
tells us1has
thatdeviation
99.7% ofofthe
68% 68-95-99.7
of the
values
are
within
std.
thescores
mean,
Only
theRule
question
changed.
deviations
from the mean advantage of
are
within
3
standard
deviations
of
the
mean.
the summary
95%we
areneed
withinto
2 standard
of the mean
andof scores
99.7%
First
find howdeviations
many standard
sheet we
99.7% are375
withis3from
standard
deviations
mean. 49.85%
49.85%
deviations
the0.15%
mean. of the
0.15%
developed.
So 375 is 3 standard deviations
𝜇 − 3𝜎
𝜇
𝜇 + 3𝜎
(25 + 25 + 25 =75) from the mean.
So the orange shaded area is 0.15%.
1000 0.15% = 1000 0.0015 =1.5 375
75
450
So, we expect about 1.5 scores to be below 375.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
To decide the percent to use, we
must find the number of standard
deviations there are between
180 and the mean (150).
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
To decide the percent to use, we
must find the number of standard
deviations there are between
180 and the mean (150).
180 − 150 = 30
180 − 150 = 30
180 − 150 = 30
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
To decide the percent to use, we
must find the number of standard
deviations there are between
180 and the mean (150).
180 − 150 = 30
30 (10+10+10) is
3 standard deviations
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
To decide the percent to use, we
must find the number of standard
deviations there are between
180 and the mean (150).
180 − 150 = 30
So, 180 is 3 standard deviations
from the mean.
30 (10+10+10) is
3 standard deviations
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
Given what
we have
learned, let’s
try to solve
this as
efficiently as
possible.
To decide the percent to use, we
must find the number of standard
deviations there are between
180 and the mean (150).
180 − 150 = 30
So, 180 is 3 standard deviations
from the mean.
First, we must
decide which
part of the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
To decide the percent to use, we
Given what
endpoints
3 standard
find theare
number
of standard
we haveWhenmust
deviations
are between
fromthere
the mean
learned, let’s deviations
180 and
mean (150).
try to solve
99.7%
ofthe
scores
this as
49.85% 49.85%
0.15%
180 − 150 = 30 0.15%
efficiently as
𝜇 − 3𝜎 So, 180 is𝜇3 standard deviations
𝜇 + 3𝜎
possible.
from the mean.
So, the
3 standard
deviation part of
the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
To decide the percent to use, we
Given what
endpoints
3 standard
find theare
number
of standard
we haveWhenmust
deviations
are between
fromthere
the mean
learned, let’s deviations
180 and
mean (150).
try to solve
99.7%
ofthe
scores
this as
49.85% 49.85%
0.15%
180 − 150 = 30 0.15%
efficiently as
𝜇 − 3𝜎 So, 180 is𝜇3 standard deviations
𝜇 + 3𝜎
possible.
from
150 the mean. 180
So, the
3 standard
deviation part of
the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
To decide the percent to use, we
Given what
endpoints
3 standard
find theare
number
of standard
we haveWhenmust
deviations
are between
fromthere
the mean
learned, let’s deviations
180 and
mean (150).
try to solve
99.7%
ofthe
scores
this as
49.85% 49.85%
0.15%
180 − 150 = 30 0.15%
efficiently as
𝜇 − 3𝜎 So, 180 is𝜇3 standard deviations
𝜇 + 3𝜎
possible.
from
150 the mean. 180
49.85% is the percent between 150 and 180.
So, the
3 standard
deviation part of
the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
To decide the percent to use, we
Given what
endpoints
3 standard
find theare
number
of standard
we haveWhenmust
deviations
are between
fromthere
the mean
learned, let’s deviations
180 and
mean (150).
try to solve
99.7%
ofthe
scores
this as
49.85% 49.85%
0.15%
180 − 150 = 30 0.15%
efficiently as
𝜇 − 3𝜎 So, 180 is𝜇3 standard deviations
𝜇 + 3𝜎
possible.
from
150 the mean. 180
49.85% is the percent between 150 and 180.
2000 49.85% = 2000 0.4985 = 997
So, the
3 standard
deviation part of
the
‘68-95-99.7 Rule’
applies.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
The distribution of scores of 2000 students on an intelligence test is a
normal distribution with mean 150 and standard deviation of 10.
How many scores do we expect to fall between 150 and 180?
To decide the percent to use, we
Given what
endpoints
3 standard
find theare
number
of standard
we haveWhenmust
deviations
are between
fromthere
the mean
learned, let’s deviations
180 and
mean (150).
try to solve
99.7%
ofthe
scores
this as
49.85% 49.85%
0.15%
180 − 150 = 30 0.15%
efficiently as
𝜇 − 3𝜎 So, 180 is𝜇3 standard deviations
𝜇 + 3𝜎
possible.
from
150 the mean. 180
49.85% is the percent between 150 and 180.
2000 49.85% = 2000 0.4985 = 997
So, the
3 standard
deviation part of
the
‘68-95-99.7 Rule’
applies.
So, we expect about 997 scores to
be between 150 and 180.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean).
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
When endpoints are 2 standard
deviations from the mean
2.5%
95% of scores
47.5%
47.5%
𝜇 − 2𝜎
𝜇
2.5%
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
When endpoints are 2 standard
deviations from the mean
Step 4. Find the percent that
95% of scores
goes with ‘below 26’.
2.5%
47.5%
𝜇 − 2𝜎
𝜇
47.5%
2.5%
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
When endpoints are 2 standard
deviations from the mean
Step 4. Find the percent that
95% of scores
goes with ‘below 26’.
26
2.5%
47.5%
𝜇 − 2𝜎
𝜇
47.5%
2.5%
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
When endpoints are 2 standard
deviations from the mean
Step 4. Find the percent that
95% of scores
goes with ‘below 26’.
26
2.5%
47.5%
𝜇 − 2𝜎
𝜇
47.5%
2.5%
𝜇 + 2𝜎
MATH 110 Sec 14-4 Lecture: The Normal Distribution
A normal distribution has a mean of 36 and a standard deviation of 5.
What percentage of values do we expect to be below 26?
Let’s use this exercise to try to get the answer while showing even less work.
Step 1. Find the difference between 26 and 36 (mean). 𝟑𝟔 − 𝟐𝟔 = 𝟏𝟎
Step 2. Express that difference in terms of standard deviations.
Standard deviation is 5 so 10 is 2 standard deviations from the mean.
Step 3. Use the percent diagram for the ‘2 standard deviation’ case.
When endpoints are 2 standard
deviations from the mean
Step 4. Find the percent that
95% of scores
goes with ‘below 26’.
26
2.5%
47.5%
47.5%
2.5%
𝜇 − 2𝜎
𝜇
𝜇 + 2𝜎
Step 5. Answer: About 2.5% of values will be below 26.