Math in Our World Section 12.4 Measures of Variation

Learning Objectives Find the range of a data set. Find the variance and standard deviation of a data set. Interpret standard deviation.

Measures of Variation In this section we will study measures of variation, which will help to describe how the data within a set vary. The three most commonly used measures of variation are range, variance, and standard deviation.

Range The range of a data set is the difference between the highest and lowest values in the set. Range = Highest value – lowest value

EXAMPLE 1 Finding the Range of a Data Set The first list below is the weights of the dogs in the first picture, and the second is the weights of the dogs in the second picture. Find the mean and range for each list. 1st: 70, 73, 58, 60 2nd: 30, 85, 40, 125, 42, 75, 60, 55

EXAMPLE 1 Finding the Range of a Data Set SOLUTION For the first list: 70, 73, 58, 60 For the second list: 30, 85, 40, 125, 42, 75, 60, 55

Variance and Standard Deviation If most of the values are similar, but there’s just one unusually high value, the range will make it look like there’s a lot more variation than there actually is. For this reason, we will next define variance and standard deviation, which are much more reliable measures of variation.

Variance and Standard Deviation Procedure for Finding the Variance and Standard Deviation Step 1 Find the mean. Step 2 Subtract the mean from each data value in the data set. Step 3 Square the differences.

Variance and Standard Deviation Procedure for Finding the Variance and Standard Deviation Step 4 Find the sum of the squares. Step 5 Divide the sum by n – 1 to get the variance, where n is the number of data values. Step 6 Take the square root of the variance to get the standard deviation.

EXAMPLE 2 Finding Variance and Standard Deviation The heights in inches of the top six scorers for the Cleveland Cavaliers during the 2008–2009 season are listed below. Find the variance and standard deviation. 80 73 87 74 83 74 SOLUTION Step 1 Find the mean height.

EXAMPLE 2 Finding Variance and Standard Deviation SOLUTION Step 2 Subtract the mean from each data value. 80 – 78.5 = 1.5 87 – 78.5 = 8.5 83 – 78.5 = 4.5 73 – 78.5 = 5.5 74 – 78.5 = 4.5 74 – 78.5 = 4.5 Data (X) X – mean (X – mean)2 80 1.5 73 – 5.5 87 8.5 74 – 4.5 83 4.5 2.25 30.25 72.25 20.25 Sum = 165.5 Step 3 Square each result. Step 4 Find the sum of the squares.

EXAMPLE 2 Finding Variance and Standard Deviation SOLUTION Step 5 Divide the sum by n – 1 to get the variance, where n is the sample size. In this case, n is 6, so n – 1 = 5. Data (X) X – mean (X – mean)2 80 1.5 73 – 5.5 87 8.5 74 – 4.5 83 4.5 2.25 30.25 72.25 20.25 Sum = 165.5 Step 6 Take the square root of the variance to get standard deviation.

Variance and Standard Deviation The variance for a data set is an approximate average of the square of the distance between each value and the mean. If X represents individual values, X is the mean and n is the number of values: The standard deviation is the square root of the variance.

EXAMPLE 3 Interpreting Standard Deviation A professor has two sections of Math 115 this semester. The 8:30 A.M. class has a mean score of 74% with a standard deviation of 3.6%. The 2 P.M. class also has a mean score of 74%, but a standard deviation of 9.2%. What can we conclude about the students’ averages in these two sections?

EXAMPLE 3 Interpreting Standard Deviation SOLUTION In relative terms, the morning class has a small standard deviation and the afternoon class has a large one. So even though they have the same mean, the classes are quite different. In the morning class, most of the students probably have scores relatively close to the mean, with few very high or very low scores. In the afternoon class, the scores vary more widely, with a lot of high scores and a lot of low scores that average out to a mean of 74%.