Center, spread, and shape of distributions | Lesson

Center, spread, and shape of distributions are also known as summary statistics (or statistics for short). These measurements are used to concisely describe data sets.

  • Center describes a typical value of in a data set. The SAT covers three measures of center: mean, median, and occasionally mode.

  • Spread describes the variation of the data. Two measures of spread are range and standard deviation.

How do I find the range and standard deviation?

On the SAT, we need to know how to find the range of a data set. While we won't be asked to calculate the standard deviation, we do need to have a sense of the relative standard deviations of two data sets.

Range

The range measures the total spread of the data; it is the difference between the maximum and minimum values.

Range = Largest value - Smallest value

(make sure the data set is organized from smallest to largest first)

A larger range indicates a greater spread in the data.

Standard deviation

Standard deviation measures the typical spread from the mean; it is the average distance between the mean and a value in the data set.

Larger standard deviations indicate a greater spread in the data.

Outliers:

The removal of an outlier is guaranteed to change the mean.

  • If a very large outlier is removed, the mean of the remaining values will decrease.

  • If a very small outlier is removed, the mean of the remaining values will increase.

Because the median is based on the middle values of a data set, an outlier does not affect the median of a data set as strongly as it affects the mean. As such, the removal of an outlier can still change the median, but that change is not guaranteed.

  • If a very large outlier is removed, the median of the remaining value will either decrease or remain the same.

  • If a very small outlier is removed, the median of the remaining value will either increase or remain the same.

Things to remember

mean = sum of values/number of values

The median is the middle value when the data are ordered from least to greatest.

  • If the number of values is odd, the median is the middle value.

  • If the number of values is even, the median is the average of the two middle values.

The mode is the most common value in a data set.

range= maximum value - minimum value

Standard deviation measures the typical spread from the mean.