3.2: Measures of Variation Version 1

Learning Objectives

• Explain measures of variation to assess how spread-out data values are.
• Calculate the range, variance, and standard deviation for grouped and ungrouped data.
• Understand the importance of variation in evaluating consistency and comparing data sets.

Measures of variation describe how data values are spread out or dispersed within a dataset. Three key measures of variation are range, variance, and standard deviation. The range is the simplest measure, calculated as the difference between the maximum and minimum values, providing a rough estimate of variability. Variance measures the average squared deviation from the mean, capturing how much data points differ from the center. The standard deviation, the square root of variance, expresses variability in the same units as the original data, making it easier to interpret. Together, these measures help in understanding the consistency and distribution of data in statistical analysis.

Range

Definition: Range

The range of a data set is a measure of variation that represents the difference between the maximum and minimum values in the set. It is calculated using the formula:

\(\text{Range} = \text{Maximum Value} - \text{Minimum Value}\)

The range provides a simple way to understand the spread of data, but it is highly sensitive to outliers since it only considers the two extreme values and ignores the distribution of the other data points.

Example \(\PageIndex{1}\)

A survey was conducted on how much money 8 university students spent on books in a semester. The recorded amounts (in dollars) were as follows:

250, 320, 275, 400, 150, 380, 290, 310

Find the range of the amount spent on books by the students.

Solution

Range = 400 - 150 = 250

Sample Variance

Sample variance is a measure of how much the values in a sample differ from the sample mean. It quantifies the spread or dispersion of the data points in a sample.

Definition: Sample Variance

The sample variance can be computed using the formula below.

\(s^2= \dfrac{n \cdot \sum x^2−(\sum x)^2}{n \cdot (n−1)} \)

where:

• \(s^2\) = sample variance
• \(n\) = number of observations in the sample
• \(x\) = individual data points
• \( \sum x\) = sum of all data points
• \( \sum x^2 \)​ = sum of the squares of all data points

Example \(\PageIndex{2}\)

A survey was conducted on how much money 8 university students spent on books in a semester. The recorded amounts (in dollars) were as follows:

250, 320, 275, 400, 150, 380, 290, 310

Find the variance of the amount spent on books by the students. Round to three decimal places.

Solution

1. Create a table with two columns. The first column lists the data values, and the second column lists the squares of the data values.

Data Values and Their Squares

x	x^2
250	62500
320	102400
275	75625
400	160000
150	22500
380	144400
290	84100
310	96100

Table \(\PageIndex{1}\): The data values and Their Squares

2. Find the sum of each column and the number of data values.

\( \sum x = 2,375\), \( \sum x^2 = 747,625\), and \( n = 8\)

3. Substitute the results of step 2 into the formula.

\(s^2= \dfrac{n \cdot \sum x^2−(\sum x)^2}{n \cdot (n−1)} \)

\(s^2= \dfrac{8 \cdot 747,625−(2375)^2}{8 \cdot 7} \)

\(s^2= \dfrac{5,981,000− 5,640,625}{56} \)

\(s^2= \dfrac{340,375}{56} \)

\(s^2 = 6078.125\)

Standard Deviation

Standard deviation measures how spread out the values in a dataset are from the mean. It indicates the average distance of data points from the mean, providing insight into the dataset's variability. A higher standard deviation means the data points are more dispersed, while a lower standard deviation indicates they are closer to the mean. It is the square root of variance and is commonly used in statistics to assess consistency, risk, or volatility in various fields.

Definition: Standard Deviation

s \( = \sqrt{s^2} = \sqrt{ Variance}\)

Example \(\PageIndex{3}\)

A survey was conducted on how much money 8 university students spent on books in a semester. The recorded amounts (in dollars) were as follows:

250, 320, 275, 400, 150, 380, 290, 310

Find the standard deviation of the amount spent on books by the students, round to three decimal places.

Solution

Use the variance from example 2 to compute the standard deviation.

s \( = \sqrt{6078.125} = 77.962\)

Example \(\PageIndex{4}\)

Studies indicate that the height of 12-year-old students in the U.S. varies based on genetics, nutrition, and other environmental factors. A classroom's height distribution can help researchers understand if the students' growth aligns with national trends.

Given the heights of 11 students (in inches):

58, 62, 65, 60, 67, 70, 64, 66, 59, 68, 63
Find the range, variance, and standard deviation of the data set. Round the answers to three decimal places.

Solution

1. Range = 70 - 58 = 12
2. Compute the variance as follows.

Create a table with two columns. The first column lists the data values, and the second column lists the squares of the data values.

Data Values and Their Squares

x	x^2
58	3,364
62	3,844
65	4,225
60	3,600
67	4,489
70	4,900
64	4,096
66	4,356
59	3,481
68	4,624
63	3,969

Table \(\PageIndex{2}\): The data values and Their Squares

Find the sum of each column and the number of data values.

\( \sum x = 702\), \( \sum x^2 = 44,948\), and \( n = 11\)

Substitute the results above into the formula for the variance.

\(s^2= \dfrac{n \cdot \sum x^2−(\sum x)^2}{n \cdot (n−1)} \)

\(s^2= \dfrac{11 \cdot 44,948−(702)^2}{11 \cdot 10} \)

\(s^2= \dfrac{494,428− 492,804}{110} \)

\(s^2= \dfrac{1624}{110} \)

\(s^2 = 14.764\)

3. Compute the standard deviation.

s \( = \sqrt{14.764} = 3.842\)

Instructions to Compute Variance and Standard Deviation Using TI-84+ Calculator

Step 1: Enter the Data

1. Press the [STAT] button.
2. Select [1: Edit...] and press [ENTER].
3. In L1 (List 1), enter each data value one by one, pressing [ENTER] after each.

Step 2: Compute the Statistics

1. Press [STAT] again.
2. Use the right arrow key to highlight [CALC].
3. Select [1: 1-Var Stats] and press [ENTER].
4. Type L1 (if your data is stored in List 1).
   • If L1 is not selected, press [2nd] and [1] to insert L1.
5. Press [ENTER] to compute the statistics.

Step 3: Locate the Standard Deviation and Variance

1. Find σx (sigma x), which represents the population standard deviation.
2. Find Sx, which represents the sample standard deviation.
   • Use the sample standard deviation Sx for our results.
   • Do not use the population standard deviation σx.
3. To find the variance, square the standard deviation.
   • Sample variance = \((S_x)^2\)

Variance and Standard Deviation for Grouped Data

The steps to find the sample variance (\(s^2\)) and standard deviation (s) for grouped data are provided below.

Step 1: Make a table as shown and find the midpoint of each class.

Titles for Grouped Data

Class	Frequency: \(f\)	Midpoint: \(X_M\)	\(f \cdot X_M\)	\(f \cdot (X_M)^2\)

Table \(\PageIndex{3}\): Headers of Grouped Frequency Distribution

Step 2: Find the sums of the second and last two columns. Substitute the sums into the formula below and solve to get the variance.

\(s^2=\dfrac{n(\sum f\cdot x_m^2)_{}-(\sum f\cdot x_m)^2}{n(n-1)}\)

Step 3: Take the square root to get the standard deviation.

\(s=\sqrt{\text{variance}} \)

Example \(\PageIndex{5}\) Variance and Standard Deviation for Grouped Data

Find the variance and standard deviation for the frequency distribution of the data.

Grouped Frequency Distribution

Class Boundaries	Frequency
5.5 - 10.5	1
10.5 - 15.5	2
15.5 - 20.5	3
20.5 - 25.5	5
25.5 - 30.5	4
30.5 - 35.5	3
35.5 - 40.5	2

Table \(\PageIndex{4}\): Grouped Frequency Distribution

Solution

Grouped Frequency Distribution

Class Boundaries	Frequency	XM	f\(\cdot\)XM	f\(\cdot\)(XM)^2
5.5 – 10.5	1	8	8	64
10.5 – 15.5	2	13	26	338
15.5 – 20.5	3	18	54	972
20.5 – 25.5	5	23	115	2645
25.5 – 30.5	4	28	112	3136
30.5 – 35.5	3	33	99	3267
35.5 – 40.5	2	38	76	2888
	20		490	13310

Table \(\PageIndex{5}\):Grouped Frequency Distribution with Added Columns

Compute the grouped variance (\(s^2\)) and standard deviation (s):

\(s^2=\dfrac{n(\sum f\cdot x_m^2)_{}-(\sum f\cdot x_m)^2}{n(n-1)}\)

\(s^2=\dfrac{20\left(13310\right)-\left(490\right)^2}{20\left(19\right)}\)

\(s^2=\dfrac{26100}{380}\approx68.68421053\)

\(s\approx\sqrt{\text{68.68421053}} \)

\(s\approx8.287593772\)

If we round to two decimal places the answers for the sample variance (\(s^2\)) and standard deviation (s) for the grouped data, we will get the following:

\(s^2\approx68.68\)

\(s\approx8.29\)

Instructions to Compute Grouped Variance and Standard Deviation Using TI-84+ Calculator

Step 1: Enter Data into the Calculator

1. Press the [STAT] button.
2. Select [1: Edit...] and press [ENTER].
3. Enter the midpoints into L1.
4. Enter the frequencies into L2.

Step 2: Calculate the Grouped Standard Deviation

1. Press [STAT].
2. Use the right arrow key to select [CALC].
3. Select [1: 1-Var Stats] and press [ENTER].
4. Make sure the screen has [List:L1] and [FreqList:L2], and then use the down arrow to select [Calculate]. (Note: Press [2nd] and 1 for L1 and Press [2nd] and 2 for L2.
5. Press [ENTER].

Step 3: Calculate the Grouped Variance

1. Find σx (sigma x), which represents the grouped population standard deviation.
2. Find Sx, which represents the grouped sample standard deviation.
   • Use the grouped sample standard deviation Sx for our results.
   • Do not use the grouped population standard deviation σx.
3. To find the grouped variance, square the standard deviation.
   • grouped variance = \((S_x)^2\)

Author

"3.2: Measures of Variation Version 1" by Alfie Swan is licensed under CC BY 4.0

Attributions

"3.2: Measures of Spread" by Kathryn Kozak is licensed under CC BY-SA 4.0