Measures of Spread

Range is a measure of spread that measures the difference between the maximum and minimum values of the data set.

Quartiles are three values that divide a data set into four equal parts. The quartiles are denoted as Q_1, Q_2, and Q_3. The second quartile Q_2, also known as the median, divides the ordered data set into two halves. ccc & Q_2& & ↑ & Lower half && Upper half a b c & d&e f g The median of the lower half is the first quartile Q_1, while the median of the upper half is the third quartile Q_3. ccc & Q_2& & ↑ & Lower half && Upper half a b c & d&e f g ↓ && ↓ Q_1&& Q_3 The first quartile is also called lower quartile, and the third quartile is also called upper quartile. To find the quartiles of a data set, the values must first be written in numerical order.

The difference between the upper and lower quartiles is the interquartile range.

The interquartile range, or IQR, of a data set is a measure of spread that measures the difference between Q_3 and Q_1, the upper and lower quartiles.

The following applet shows how to find the IQR of different data sets.

The mean absolute deviation (MAD) is a measure of the spread of a data set that measures how much the data elements differ from the mean. The mean absolute deviation is the average distance between each data value and the mean.

Calculating the MAD involves determining the absolute difference between every data point and the mean, followed by averaging these absolute differences. The applet below calculates the mean absolute deviation for the data set on the number line. Move the points around to change the data.

A large MAD value indicates that data points deviate considerably from the mean — that is, there is significant variation within the data set. The mean absolute deviation is less sensitive to outliers compared to standard deviation and variance because it calculates the average of the absolute differences.

The standard deviation is a measure of spread of a data set that measures how much the data elements differ from the mean. The standard deviation, often represented by the Greek letter σ (sigma), is calculated by taking the square root of the variance of the data set. Let x_1, x_2,..., x_n be the data values in a set and x their mean. σ = sqrt((x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n) The applet below calculates the standard deviation for the data set on the number line. Move the points around to change the data.

As shown, finding the standard deviation involves calculating the average of the squared differences between each data point and the mean, and then taking the square root of that average. The sum of squares can also be written in sigma notation. σ = sqrt(∑_(i=1)^n(x_i-x)^2/n)

Standard deviation is sensitive to outliers because of the squaring of differences. It is commonly used when analyzing a data set that exhibits a normal distribution.

The variance is a measure of spread of a set of data that measures how much the data elements deviate from the mean. Mathematically, the variance is the average of the squares of the difference between each data value x_i and the mean x. (x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n The variance is the square of the standard deviation σ, so it is usually denoted as σ ^2. σ = sqrt((x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n) ⇕ σ ^2 = (x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n The applet below calculates the variance in the data set on the number line. Points can be moved to change the data.

The standard deviation, σ, of a data set is calculated using the following formula. σ = sqrt((x_1 - x)^2 + (x_2 - x)^2 + ... + (x_n - x)^2/n) In this formula, n is the number of values in the data set and x is the mean of the data set. Performing this calculation in one step makes for a convoluted expression. Therefore, it is best divided into a few, smaller steps. Consider the following data set as an example. 1, 5, 3, 4, 5, 12 Follow these steps when finding the standard deviation.