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I can interpret bell curves and use z-scores
Warm Up
1. For what value(s) of 𝜃 is 𝑠𝑖𝑛𝜃 = .2756?
2. The average IQ in a town is 112 with a
standard deviation of 11.
a) What percent of the town has an IQ
between 112 and 123?
b) What percent of the population has an IQ
below 90 or above 134?
1. For what value(s) of 𝜃 is 𝑠𝑖𝑛𝜃 = .2756?
𝑠𝑖𝑛−1 .2756 = 16°
Sine is also positive in quadrant….
Two
180 − 𝜃 = 𝑥
180 − 16 = 𝑥
𝑥 = 164
2. The average IQ in a town is 112 with a
standard deviation of 11.
a) What percent of the town has an IQ
between 112 and 123?
34%
90
101
112 123
134
2. The average IQ in a town is 112 with a
standard deviation of 11.
b) What percent of the population has an IQ
below 90 or above 134?
100 − 95 =
5%
90
101
112 123
134
Homework Questions
Five hundred values are normally distributed with a
mean of 125 and a standard deviation of 10.
b. What percent of the values is in the interval
100 - 150, to the nearest percent?
105
115 125
135
145
2nd VARS
2: normalcdf(
Lower: 100
Upper:150
𝜇: 125
𝜎: 10
Enter, Enter
.987  98.7%
The average height of 6th graders at Beverly Cleary
Middle School is 60 inches with a standard
deviation of 4 inches.
What percent of students are between 56 and 61
inches?
44%
What percent of students are between 50 and 60
inches?
49%
What percent of students are taller than 62
inches?
31%
Period 1
Is a 76 the same in both classes?
Period 4
Because means and standard deviations vary, it
is difficult to compare data. To normalize data
for easy comparison, we use z-scores.
A z-score is a way of measuring the distance of a
value from the mean.
The Bell Curve can be redrawn to incorporate zscores
-2
-1
0
1
2
Positive z-score values are above the mean;
Negative z-score values are below the mean
• Z-score Distributions will always have a mean of zero
and a standard deviation of one
• Does not change the shape of the original distribution
and does not change the location of any individual
score relative to others in the distribution
𝑥 − 𝑚𝑒𝑎𝑛
𝑧=
𝜎
Period 1 Test Score
The mean is 70 the standard deviation is 3. Find
the z-score for 76
76 − 70
𝑧=
3
𝑧=2
𝑥 − 𝑚𝑒𝑎𝑛
𝑧=
𝜎
Period 4 Test Score
The mean is 70 the standard deviation is 12.
Find the z-score for 76
76 − 70
𝑧=
12
𝑧 = 0.5
Compare our two test scores of 76.
Period 1: 𝑧 = 2
Period 2: z = 0.5
Which student did better in comparison to
his/her classmates?
-2
-1
0
1
2
1) Mean of 72, std dev of 7, score 81
𝑧 = 1.29
2) Mean of 18, std dev of .5, score 16.75
𝑧 = −2.5
3) Mean of 105, std dev of 22, score 97
𝑧 = −.36
4) Mean of 35, std dev of 6, score 50
𝑧 = 2.5
The annual per person consumption of apples in
the U.S. is 16 pounds of apples with a standard
deviation of 4 pounds.
Find the z-score for an annual per-person
consumption of 22 pounds.
What is the probably that a randomly selected
person eats less than 22 pounds of apples?
Lexi took the ACT and received a score of 33.
The ACT has a mean of 27 and a standard
deviation of 1.6
Ryan took the SAT and earned a 2300. The SAT
has a mean of 2100 and a standard deviation of
172.
Who did better on their college entrance exam?
Worksheet