4.4: Measures of Variation

Range

Video Introduction to Variation (2 mins 12 secs – CC).

We saw already in the previous section how to measure the center of a quantitative data set using the mean, median or mode, but what about the spread? We will start off with some exploration and build up a set of tools to allow us to give some numeric summary for a measure of the spread.

Consider these three sets of quiz scores:

All three of these sets of data have a mean of 5 and median of 5 (verify this on your own), yet the scores for each section are clearly quite different. In section A, everyone had the same score; in section B half the class got no points and the other half got a perfect score, assuming this was a 10-point quiz. Section C was not as consistent as section A, but not as widely varied as section B. A numeric summary of the center doesn’t tell us this about the data. From the numeric summary of center they all would appear equivalent if we didn’t have the data itself in front of us. We need some way to describe how the data varies about that center.

There are several ways to measure this “spread” of the data. The first is the simplest and is called the range.

In each of the sections from above we saw that the median was 5 and now have a tool to help describe the spread by using the range. Does the range do a good job of describing the spread?

Not exactly. In the last example, range seems to be revealing how spread out the data is. However, suppose we add a fourth section, Section D, with scores 0 5 5 5 5 5 5 5 5 10.

This section also has a mean and median of 5. The range is 10, yet this data set is quite different than Section B. The range doesn’t take into account how close all the data points are to the center. To better illuminate the differences, we’ll have to turn to more sophisticated measures of variation. We call it the standard deviation – the formula looks complicated, but we will walk through the details and how to calculate it by hand (typically software is used and can be done on most handheld calculators and spreadsheet programs).

Standard Deviation and Variance

If x is a number, then the difference “x-mean” is called its deviation. What we want to find is some way to describe the deviations of an entire data set. The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ. The formula for finding the standard deviation is given below.

Using the data from section D, we could compute for each data value the difference between the data value and the mean:
 

We would like to get an idea of the “average” deviation from the mean, but if we find the average of the values in the second column the negative and positive values cancel each other out (this will always happen), so to prevent this we square every value in the second column:
 

We then add the squared deviations up to get 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 25 = 50.  Ordinarily we would then divide by the number of scores, n, (in this case, 10) to find the mean of the deviations.  But we only do this if the data set represents a population; if the data set represents a sample (as it almost always does), we instead divide by n – 1 (in this case,  10 – 1 = 9). The reason for doing so is that it provides a better estimator to the actual standard deviation of a population (a proof of this is beyond the scope of our work).

So in our example, we would have 50/10 = 5 if section D represents a population and 50/9 = about 5.56 if section D represents a sample.  These values (5 and 5.56) are called, respectively, the population variance and the sample variance for section D.

Variance can be a useful statistical concept, but note that the units of variance in this instance would be points-squared since we squared all of the deviations.  What are points-squared?  Good question.  We would rather deal with the units we started with (points in this case), so to convert back we take the square root and get:

$$\begin{array}{l}\text{population standard deviation, }\sigma =\sqrt{\frac{50}{10}}=\sqrt{5}\approx 2.2\hfill \\ \text{or}\hfill \\ \text{sample standard deviation, s}=\sqrt{\frac{50}{9}}\approx 2.4\hfill \end{array}$$

If we are unsure whether the data set is a sample or a population, we will usually assume it is a sample, and we will round answers to one more decimal place than the original data, as we have done above.

Doing these calculations by hand can be tedious, but you have to be careful that when you use software to compute the standard deviation that it is most likely calculating the standard deviation for a sample.

The work we have done to find the standard deviation can be long and labor intensive for large data sets (even for calculating the mean and median). It is best to use software to do these calculations when it is available. Applications like Excel and Google Sheets both have functions built into them to help you with this. When using make sure you know if you are finding the standard deviation for a population or sample though as we have already seen the formulas are slightly different depending on where your data is coming from.

Excel STDEV.S or STDEV.P

Google Sheets STDEV or STDEVP

Five-number summary

Where standard deviation is a measure of variation based on the mean, quartiles are based on the median.

While quartiles are not a 1-number summary of variation like standard deviation, the quartiles are used with the median, minimum, and maximum values to form a 5 number summary of the data.

The five number summary gives a numeric description for the data set. We can go further by quantifying a spread about the median with the interquartile range and define a way to classify data that are outliers. We consider a point an outlier when it is substantially above Q3 or substantially below Q1. In the definition below we quantify what substantial means.

To find the first quartile, we need to find the data value so that 25% of the data is below it. If n is the number of data values, we compute a locator by finding 25% of n. If this locator is a decimal value, we round up, and find the data value in that position. If the locator is a whole number, we find the mean of the data value in that position and the next data value. This is identical to the process we used to find the median, except we use 25% of the data values rather than half the data values as the locator.

The goal is to guarantee that we select the smallest data value for which 25% of the data falls below the data point in that L+ position. If we had rounded down we would have less than 25% of the data falling below L+.

Instructions on finding quartiles vary, but the general goal is to determine a value that helps measure the spread of the data. Examples should help make this clearer.

Of course, with a relatively small data set, finding a five-number summary is a bit silly, since the summary contains almost as many values as the original data set. In the Try it Now below you will have a larger data set.

In the next example below we will find the five number summary when the data is given on a frequency table.

Box-Plot

The 5 number summary divides the data into four intervals, each of which will contain about 25% of the data. For visualizing that summary, there is a graphical representation of a 5-number summary called a box plot, or box and whisker graph.

Notice how the boxplot allows us to visualize the differences in the spread of the data on the quartiles instead of just having the range given to us.

Applications of Box Plots

Box plots are particularly useful for comparing data from two populations.

Below is an example from Khanacademy. Make sure you pause the video and work the problem yourself. When you compare the solutions you will see he uses a different method for finding the quartiles – this method will give you the same results as you would get with the method shown above. Just keep in mind that we are using the locator method (.25n) and (.75n) with rounding up for decimals to find Q1 and Q3.

Box and Whisker Plot (13 mins 14 secs – CC)

Exercises

1. Determine the range for the set of scores: 10, 8, 5, 4, 6, 3, 5, 6, 7.
   
   Answer (click to Show/Hide)

   Range: $10-3=7$

2. The temperature highs (in Fahrenheit) for the first ten days in May 2021 for the temperature in Tucson are given below. Round all answers to two decimal places.
   

   95°	92°	86°	91°	96°	100°
   96°	92°	93°	94°	94°	97°

   1. Find the mean, median, and mode for the data set.
   2. Find the range, variance, and standard deviation for the data set. Treat the data as if it were a sample.
   3. Find the five-number summary for the data set.
   4. Draw a box plot for the data set.
   Answer (click to Show/Hide)

   1. Mean 93.83, Median 94, Mode 92, 94, 96
   2. Range 14, Standard Deviation 3.51, Variance 12.33
   3. 86, 92, 94, 96, 100
   4. 
3. A sample of 30 distance scores measured in yards has a mean of 7, a variance of 16, and a standard deviation of 4.
   1. You want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. What are the new mean, median, variance, and standard deviation?
   2. You then decide that you only want to look at the distance past a certain point. Thus, after multiplying the original scores by 3, you decide to subtract 4 feet from each of the scores. Now what are the new mean, median, variance, and standard deviation?
   Answer (click to Show/Hide)

   1. The mean is 21, standard deviation is 12, variance is 144.
   2. The mean is 17, standard deviation is 12, variance is 144. Notice the spread did not change when subtracting 4 from each value.
4. Make up three data sets with 5 numbers each that have the same mean but different standard deviations.
5. Would the variance of 10, 12, 17, 20, 25, 27, 42, and 45 be larger if the numbers represented a population or a sample?
   
   Answer (click to Show/Hide)

   The variance would smaller for a population as we divide by a larger number in the denominator when working data that represents the population.

6. You recorded the time in seconds it took for 8 participants to solve a puzzle. The times were:  15.2, 18.8, 19.3, 19.7, 20.2, 21.8, 22.1, 29.4. Find the variance and standard deviation for the times (assume the data comes from a sample of all puzzle solvers).
   
   Answer (click to Show/Hide)

   Standard Deviation 4.07, Variance 16.55

7. You recorded the time in seconds it took for 8 participants to solve a puzzle. These times appear below. However, when the data was entered into the statistical program, the score that was supposed to be 22.1 was entered as 21.2. Scores: 15.2, 18.8, 19.3, 19.7, 20.2, 21.8, 22.1, 29.4
   1. Which of these measures of central tendency will change when you correct the recording error (mean, median, and/or mode)?
   2. Which measures of variability (range, standard deviation, variance) will change when you correct the recording error?
   Answer (click to Show/Hide)

   1. The Mean would change.
   2. The standard deviation and variance would change.
8. A random sample of students were selected from a local college and their age was recorded. The ages for the 25 students were: 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 25, 26, 29, 30, 32, 38, 51. Create a boxplot of the ages.
   
   Answer (click to Show/Hide)

   The five number summary: 17, 19, 20, 25, 51

   

9. The box plot below shows salaries for Actuaries and CPAs. Kendra makes the median salary for an Actuary. Kelsey makes the first quartile salary for a CPA. Who makes more money? How much more?

   

   Answer (click to Show/Hide)

   The median salary for Actuaries is 90,000. The first quartile for CPA salaries is 40,000. Kendra made $\mathrm{\$}90,000-\mathrm{\$}40,000=\mathrm{\$}50,000$ more than Kelsey.

10. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. 

    Researcher One: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34

    Researcher Two: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29

    Create a comparative set of boxplots of the data for the researchers.

    Answer (click to Show/Hide)

    The five number summary Researcher One: 1, 15, 24, 31.5, 47

    The five number summary Researcher Two: 2, 16, 22, 30, 44

    

11. The following box plot shows the estimated percent of U.S. population below a given age for 2019 (from census.gov).
    
    1. Can we say children (17 and under) make up 25% of the population?
    2. What percent of the population is between 19 and 58?
    3. If the total population is estimated to be 328,239,523, then approximately how many people are between the ages 19 and 39?
    4. 11.5% are age 65 and over. Approximately what percentage of the population are working adults (age 18 to age 65)?
    Answer (click to Show/Hide)

    The five number summary: 0, 19, 39, 58, 105

    1. No. The first quartile ends at 19, so less than 25% of the population is below the age of 18.
    2. The first and third quartile are 19 and 58 respectively, so 50% of the population will be between 19 and 58.
    3. The first quartile is at age 19 and the median is at age 39. This means 25% of the population is between those ages and gives $328,239,523\cdot \left(0.25\right)=82,059,880$ for the population between those ages.
    4. If we consider the first quartile as being really close to 18 and say 25% are below the age of 18, then we can approximate this percentage as $100\mathrm{\%}-11.5\mathrm{\%}-25\mathrm{\%}=63.5\mathrm{\%}$. This result may be slightly off.

Attributions

This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.

This page contains modified content from “OpenStax Introductory Statistics” by Barbara Illowsky, Susan Dean. Licensed under CC BY 4.0.

The data from Example 11 is from van Vliet, P.K. and Gupta, J.M. (1973) Sodium bicarbonate in idiopathic respiratory distress syndrome. Arch. Disease in Childhood, 48, 249–255. As quoted on http://openlearn.open.ac.uk/mod/oucontent/view.php?id=398296§ion=1.1.3

This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.