1. 12-4 Standard Deviation
Hubarth
Algebra II

2. Ex. 1 Real-World Connection
Statisticians use several measures of variation to describe how the data in a data set are
spread out.
The range of a set of data is the difference between the greatest and least values. The
interquartile range is the difference between the third and first quartile.
Ex. 1 Real-World Connection
There are 9 members of the Community Youth Leadership Board. Find the range and interquartile range of their ages: 22, 16, 24, 17, 16, 25, 20, 19, 26.
greatest value – least value = 26 – Find the range.
= 10
median Find the median.
Q1 = = 16.5     Q3 = = Find Q1 and Q3.
( ) 2
( )
Q3 – Q1 = 24.5 – Find the interquartile range.
= 8
The range is 10 years. The interquartile range is 8 years.

3. Another measure of variation is the standard deviation, a measure of how much the values
in a data set vary, or deviate, from the mean. The Greek letter 𝜎 (sigma) represents standard
deviation.

4. Ex. 2 Finding the Standard Deviation
Find the mean and the standard deviation for the values 78.2, 90.5, 98.1, 93.7, 94.5.
= = 91   Find the mean.
( ) 5 x
Organize the next
steps in a table. – – x
x – x
(x – x)2
 = Find the standard deviation.
(x – x)2 n
234.04 5 =
The mean is 91, and the standard deviation is about 6.8.

5. Ex. 3 Real-World Connection
Use the data to find the mean and standard deviation for daily energy demand on the weekends only.
S M T W TH F S
Step 1: Use the STAT feature
to enter the data as L1.
Step 2: Use the CALC menu of
STAT to access the
1-Var Stats option.
The mean is about 36.1 MWh;
The mean is 𝑥 .
the standard deviation
is about 3.6 MWh.
The standard
deviation is x.

6. Ex. 4 Real-World Connection
The number of points that Darden scored in each of 11 basketball games is listed below. Within how many standard deviations of the mean do all of the values fall? What can Darden’s coach do with this information?
8, 12, 13, 10, 7, 5, 10, 9, 13, 11, 8
Step 2:  Mark off intervals of 2.5 on
either side of the mean.
Step 1:  Draw a number line. Plot the
data values and the mean.
All the values fall within 2 standard
deviations of the mean.
The coach can expect that it will be very likely
that his score in the next game will
be within 5 points of his mean score
of 10 points.

7. The z-score is the number of standard deviations that a value is from the mean.
Ex. 5 Finding the Z-Score
A set of values has a mean of 22 and a standard deviation of 5. Find the z-score for a value of 30.
z-score =
value – mean
standard deviation
= Substitute.
30 – 22 5
= Simplify. 8 5
= 1.6

8. Practice
1. Seventeen women qualified for the 2002 U.S. Women’s Alpine Ski Team. Find the range
and the interquartile range of their ages:
24, 30, 29, 21, 22, 22, 28, 21, 16, 17, 25, 22, 21, 18, 19, 18, 19
Range: 14, interquartile range: 6
2. Find the mean and standard deviation for the values: 50, 60, 70, 80, 80, 90, 100, 110
𝑥 =80, 𝜎=18.7
3. Find the mean and standard deviation for this data set: 2mm, 3mm, 4mm, 6mm, 7mm, 9mm,
10mm, 12mm, 13mm, 14mm.
𝑥 =8, 𝜎≈4.0
4. A set of values has a mean of 85 and a standard deviation of 6. Find the value that has a
z-score of 2.5.
100