In the next activity, you will be exploring data and using the measures of center and measures of spread to describe the data. This section will introduce you to variability, which is a measure of how dispersed (spread out) the data are. You’ll learn to recognize variability in histograms and dotplots by using visual clues. You’ll also learn how to calculate measures of variability including standard deviation, variance, and range.

Comparing Variability

The variability of a dataset is often referred to as the spread of a dataset. We can visually assess variability using graphical displays such as histograms and dotplots. When answering Questions 1 – 3 below, consider whether the data appears to be more spread out from the center (greater variability) or more clustered toward the center (less variability).

Comparing Variability Using Histograms

Recall that a histogram visualizes the distribution of a quantitative variable by displaying rectangular bars representing the frequencies (height of the bar) for intervals of data values called bins (width of the bar). Variability can be judged from a histogram by examining the distance of the bars from the statistical center (mean or median) of the graph. If the variability is high, equally sized or taller bars will appear away from the center of the graph. It the variability is low, the data will appear clustered around the center.

The images below show distributions of two different datasets using histograms. The first histogram displays the distribution of responses given by parents of thirteen year old children to the question, “how much allowance do you give weekly?” The second is a distribution of the heights in inches of 31 thirteen year old boys attending the same middle-school. Use these histograms to answer Question 1 below.

Using Dotplots to Visually Compare Variability

A dotplot indicates the variability of the data or the extent to which each observation differs from other observations. It can be easier to visualize variability using a dotplot than using a histogram because of the individual observations visible in the dotplot. Use the side-by-side dotplots in the image below to answer Questions 2 and 3.

Ten customers rated four different smartphone apps. The customer ratings for the four different apps are shown in the following dotplots. The mean for each app is equal to 3. Even though the mean, [latex]\bar{x}[/latex], is the same for each app, the dotplots for each app look very different.

Using Technology to Obtain Descriptive Statistics

Let’s go to the technology now and recall how to load a dataset in order to describe and explore it.

For Questions 4 and 5, recall the sleep study in which you investigated whether college students’ chronotypes tend to be larks (morning people) or owls (night people) by examining graphical representations of the data. Let’s use the dataset from that study again here.

Measures of Variability

In statistics, we are particularly interested in understanding how data are distributed and where each observation is in reference to the mean. How spread out a set of observations are is called variability (also called spread or dispersion). In the remainder of this section, we will focus on three measures of spread: standard deviation, variance, and range.

Standard deviation is a measure of how spread out observations are from the mean. The symbol we use to denote standard deviation differs depending on whether we are discussing a sample or a population. We use the Greek letter [latex]\sigma[/latex] (sigma) to denote the standard deviation of a population of observations. We use the Latin letter [latex]s[/latex] to denote the standard deviation of a sample of observations.

The following formulas are used to calculate the standard deviation of a population and a sample:

Standard deviation of a population: [latex]\sigma = \sqrt{\dfrac{\sum \left(x-\mu\right)^2}{n}}[/latex]

Standard deviation of a sample: [latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex]

The following steps can be applied to calculate a standard deviation by hand.

1. Calculate the mean of the population or sample.
2. Take the difference between each data value and the mean. Then square each difference.
3. Add up all the squared differences
4. Divide by either the total number of observations in the case of a population or by 1 fewer than the total in the case of a sample.
5. Take the square root of the result of the division in step 4.

Here is a breakdown of the formula for standard deviation of a sample, [latex]s[/latex].

[latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex]

• The distance from each observation to the mean is known as a deviation from the mean and is expressed as [latex]\left(x-\bar{x}\right)[/latex]
• The deviations from the mean are squared in the formula because some observations are above the mean, thus [latex]\left(x-\bar{x}\right)>0[/latex] (the difference is positive), and some observations are below the mean, thus [latex]\left(x-\bar{x}\right)<0[/latex] (the difference is negative). Squaring ensures the differences will each be expressed as positive distances and won’t cancel each other out when summed up.
• The [latex]\sum[/latex] symbol sums up the squared deviations for all [latex]n[/latex] observations.
  
• The denominator in the formula for a sample standard deviation is [latex]\left(n-1\right)[/latex] rather than [latex]n[/latex] as in the formula for the population standard deviation.
  • Why do we divide by 1 fewer than the sample size, [latex]\left(n-1\right)[/latex]?
    • The answer to that is complicated but here are some ideas that may help