Download Standard Deviation

Objectives
The student will be able to:
•
find the variance of a data set.
•
find the standard deviation of a
data set.
Interquartile range and
outliers



The interquartile range is
Q3  Q1
Outliers are numbers far from the others.
Technically ,outliers are data that fall
beyond these:
Q3  (1.5  IQR)
Q1  (1.5  IQR)
outlier

A data point that is distinctly separate from
the rest of the data. A piece of data that is
far away from the other pieces is an outlier.
One definition of outlier is any data point
more than 1.5 times interquartile range
(IQR) below the first quartile or above the
third quartile
Interquartile range (IQR)
Example:
For the data 2, 5, 6, 9, 12, we have
the following summary:
first quartile = 3.5
median = 6
 third quartile = 10.5
IQR = 10.5 – 3.5 = 7, so 1.5(IQR) = 10.5.
 To determine if there are outliers we must consider
the numbers that are 1.5·IQR or 10.5 beyond the
quartiles.
 first quartile – 1.5·IQR = 3.5 – 10.5 = –7
third
quartile + 1.5·IQR = 10.5 + 10.5 = 21

Variance
Variance is the average
squared deviation from the
mean of a set of data. It is
used to find the standard
deviation.
Variance
1.
Find the mean of the data.
Hint – mean is the average so add up the
values and divide by the number of items.
2. Subtract the mean from each value – the
result is called the deviation from the mean.
3. Square each deviation of the mean.
4. Find the sum of the squares.
5. Divide the total by the number of items.
Variance Formula
The variance formula includes the
Sigma Notation, which represents the
sum of all the items to the right of
2
Sigma.
(x   )

n
Mean is represented by
the number of items.
 and n is
Standard Deviation
Standard Deviation shows the
variation in data. If the data is close
together, the standard deviation will
be small. If the data is spread out, the
standard deviation will be large.
Standard Deviation is often denoted
by the lowercase Greek letter sigma,

.
Standard Deviation
Standard Deviation for a sample of
the population is S on calculator.
Standard Deviation for an entire
population is

The mean is denoted by
on the calculator.
X
Standard Deviation
Find the variance.
a) Find the mean of the data.
b) Subtract the mean from each value.
c) Square each deviation of the mean.
d) Find the sum of the squares.
e) Divide the total by the number of
items.
Take the square root of the variance.
Standard Deviation Formula
The standard deviation formula can be
represented using Sigma Notation:

2
(
x


)

n
Notice the standard deviation formula
is the square root of the variance.
Find the variance and
standard deviation
The math test scores of five students
are: 92,88,80,68 and 52.
1) Find the mean: (92+88+80+68+52)/5 = 76.
2) Find the deviation from the mean:
92-76=16
88-76=12
80-76=4
68-76= -8
52-76= -24
Find the variance and
standard deviation
The math test scores of five
students are: 92,88,80,68 and 52.
3) Square the deviation from the
mean: (16) 2  256
(12)  144
2
(4)  16
2
(8)  64
2
(24)  576
2
Find the variance and
standard deviation
The math test scores of five students
are: 92,88,80,68 and 52.
4) Find the sum of the squares of the
deviation from the mean:
256+144+16+64+576= 1056
5) Divide by the number of data
items to find the variance:
1056/5 = 211.2
Find the variance and
standard deviation
The math test scores of five students
are: 92,88,80,68 and 52.
6) Find the square root of the
variance: 211.2  14.53
Thus the standard deviation of
the test scores is 14.53.
Calculator


Go to “stat” edit and enter the data
Go to “stat, calc 1, L1
Standard Deviation
A different math class took the
same test with these five test
scores: 92,92,92,52,52.
Find the standard deviation for
this class.
Hint:
1. Find the mean of the data.
2. Subtract the mean from each value
– called the deviation from the
mean.
3. Square each deviation of the mean.
4. Find the sum of the squares.
5. Divide the total by the number of
items – result is the variance.
6. Take the square root of the
variance – result is the standard
deviation.
Solve:
A different math class took the
same test with these five test
scores: 92,92,92,52,52.
Find the standard deviation for this
class.
Answer Now
The math test scores of five students
are: 92,92,92,52 and 52.
1) Find the mean: (92+92+92+52+52)/5 = 76
2) Find the deviation from the mean:
92-76=16 92-76=16 92-76=16
52-76= -24
52-76= -24
3) Square the deviation from the mean:
(16)2  256(16)2  256(16) 2  256
  
4) Find the sum of the squares:
256+256+256+576+576= 1920
The math test scores of five
students are: 92,92,92,52 and 52.
5) Divide the sum of the squares
by the number of items :
1920/5 = 384 variance
6) Find the square root of the variance:
384  19.6
Thus the standard deviation of the
second set of test scores is 19.6.
Analyzing the data:
Consider both sets of scores. Both
classes have the same mean, 76.
However, each class does not have the
same scores. Thus we use the standard
deviation to show the variation in the
scores. With a standard variation of
14.53 for the first class and 19.6 for the
second class, what does this tell us?
Answer Now
Analyzing the data:
Class A: 92,88,80,68,52
Class B: 92,92,92,52,52
With a standard variation of 14.53
for the first class and 19.6 for the
second class, the scores from the
second class would be more spread
out than the scores in the second
class.
Analyzing the data:
Class A: 92,88,80,68,52
Class B: 92,92,92,52,52
Class C: 77,76,76,76,75
Estimate the standard deviation for Class C.
a) Standard deviation will be less than 14.53.
b) Standard deviation will be greater than 19.6.
c) Standard deviation will be between 14.53
and 19.6.
d) Can not make an estimate of the standard
deviation.
Answer Now
Analyzing the data:
Class A: 92,88,80,68,52
Class B: 92,92,92,52,52
Class C: 77,76,76,76,75
Estimate the standard deviation for Class C.
a) Standard deviation will be less than 14.53.
b) Standard deviation will be greater than 19.6.
c) Standard deviation will be between 14.53
and 19.6
d) Can not make an estimate if the standard
deviation.
Answer: A
The scores in class C have the same
mean of 76 as the other two classes.
However, the scores in Class C are all
much closer to the mean than the other
classes so the standard deviation will be
smaller than for the other classes.
Summary:
As we have seen, standard deviation
measures the dispersion of data.
The greater the value of the
standard deviation, the further the
data tend to be dispersed from the
mean.
The bell curve which represents a
normal distribution of data shows
what standard deviation represents.
X
One standard deviation away from the mean ( X ) in
either direction on the horizontal axis accounts for
around 68.2 percent of the data. Two standard
deviations away from the mean accounts for roughly
95 percent of the data with three standard
Use the curve to fill in values underneath:
2.2
X
Values go under here for each problem.
68.2%
1.7+ .5 +.1 = 2.3%
Ex: Mean = 16 and standard deviation = 6:
2.2
-2 1
4
10
X
16
68.2%
22
28
34
1.7+ .5 +.1 = 2.3%
Mean = 64, standard deviation is 3.
What % is less than 58?
2.2
2.2
58
61
64
67
70
Less than 58 is 1.7 + .5 +.1 = 2.3%
Mean = 84, standard deviation is 4. How
many of 31 days are above 90?
2.2
2.2
76
80
84
88
90
Above 90 is 4.4 +1.7 + .5 +.1 = 6.7%
Mean = 84, standard deviation is 4. How
many of 31 days are above 90?
2.2
2.2
76
80
84
88
Above 90 is 6.7%
31(.067)=2.077, so 2 days
90
Normal Distribution


Reference
sheet
Re-draw fo
problems
Z-Scores above the mean is
positive, below is negative

Formula:

z
xx

Given the mean is
60 and standard
deviation is 6,
what is the
z- score of 48?
Z-Scores: how many standard
deviations from the mean

Formula:
xx

48  60
 2
6

Given the mean is
60 and standard
deviation is 6,
what is the
z- score of 48?
Z-Scores: how many standard
deviations from the mean

Formula:
z
xx


Given the mean is
160 and 186 has
a z-score of 2,
find the standard
deviation.
Z-Scores: how many standard
deviations from the mean

Formula:
z
2
xx

186 160

2  26
  13

Given the mean is
160 and 186 has
a z-score of 2,
find the standard
deviation.
Summary:

Percents are
found on the
curve.


Some problems
use the % to
answer for a
number
Some problems
ask what % are
certain values
50 % are above 16, 50% are below 16.
15.9% are above 22, etc.
2.2
-2 1
4
10
16
68.2%
22
28
34
1.7+ .5 +.1 = 2.3%
precentiles
2.2
.1%
2.2
2.3% 15.9% 50% 84.1% 97.7% 99.9%
Vocabulary


Survey-collecting
and analyzing
data.
Reliable data is
based on random
samples with no
bias.


Bias-choosing
people that may
have a specific
opinion
Un-baisedrandomly selected
Experiments

Observationalsimply watching
what happens
without
influencing the
outcome.


Choose, analyze,
measure.
Controlled
experiment
involves changing
some aspect of
the study