12.4 – Standard Deviation

Measures of Variation

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 quartiles

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 – 16 Find the range.

= 10

Q3 – Q1 = 24.5 – 16.5 Find the interquartile range.= 8

The range is 10 years. The interquartile range is 8 years.

median Find the median.

16 16 17 19 20 22 24 25 26

Q1 = = 16.5     Q3 = = 24.5 Find Q1 and Q3.(16 + 17)

2(24 + 25)

2

Example

More Measures of Variation

Standard deviation is a measure of how each value in a data set varies or deviates from the mean

2. Find the difference between each value and the mean:

3. Square the difference:

5. Take the square root to find the standard deviation

4. Find the average (mean) of these squares:

1. Find the mean of the set of data:

Steps to Finding Standard Deviation

x

xx 2)( xx

n

xx 2)(

n

xx 2)(

Standard Deviation

Find the mean and the standard deviation for the values 78.2, 90.5,

98.1, 93.7, 94.5.

The mean is 91, and the standard deviation is about 6.8.

234.045

= 6.8

= = 91   Find the mean.(78.2 + 90.5 + 98.1 +93.7 +94.5)5

x

= Find the standard deviation.

(x – x)2

n

Organize the nextsteps in a table.78.2 91 –12.8 163.84

90.5 91 –0.5 .2598.1 91 7.1 50.4193.7 91 2.7 7.2994.5 91 3.5 12.25

x

x

x – x

(x – x)2

Let’s Try One – No Calculator!

Find the mean and the standard deviation for the values 9, 4, 5, 6

  Find the mean. x

= Find the standard deviation.

(x – x)2

n

Organize the nextsteps in a table.

x

x

x – x

(x – x)2

Let’s Try One – No Calculator

Find the mean and the standard deviation for the values 9, 4, 5, 6

The mean is 6, and the standard deviation is about 1.87.

= = 6  Find the mean.(9+4+5+6)4

x

= Find the standard deviation.

(x – x)2

n

Organize the nextsteps in a table.9 6 3 9

4 6 -2 45 6 -1 16 6 0 0

sum 14

x

x

x – x

(x – x)2

87.12

14

4

56

4

4

4

14

More Measures of Variation

Z-Score: The Z-Score is the number of standard deviations that a value is from the mean.

A set of values has a mean of 22 and a standard deviation of 3. Find the

z-score for a value of 24.

Z-Score

= 0.6

= Simplify.23

= Substitute.24 – 22

3

z-score = value – mean standard deviation

A set of values has a mean of 34 and a standard deviation of 4. Find the

z-score for a value of 26.

Z-Score

= -2

= Simplify.-84

= Substitute.26 – 34

4

z-score = value – mean standard deviation

Standard DeviationUse the data to find the mean and standard deviation for daily

energy demands on the weekends only.

Step 1: Use the STAT feature to enter the data as L1.

S M T W Th F S53 52 47 47 50 39

33 40 41 44 47 49 4339 47 49 54 53 46 3633 45 45 42 43 39 3333 40 40 41 42

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 standarddeviation is x.