Measures of Central Tendency
Measures of Central Tendency
Measures of central tendency identify the center, or most typical, value within a data set. There are three such central tendencies—mean, median, and mode—that describe the “center” of the data in different ways. The mean (μ) is the arithmetic average and is found by dividing the sum of all measurements by the number of measurements.
L\(\mu = \frac{x_1\ +\ x_2\ +\ …x_N}{N}\)
Find the mean of the following data set: {75, 62, 78, 92, 83, 90}
\(\mu = \frac{75\ +\ 62\ + 78\ +\ 92\ +\ 83\ +\ 90}{6} = \frac{480}{6} = 80\)
The median divides a set into two equal halves. To calculate the median, place the data set in ascending order. The median is the middle value in a set of data.
Find the median of the following data set: {2, 15, 16, 8, 21, 13, 4}
Place the data in ascending order: {2, 4, 8, 13, 15, 16, 21}
median = 13
If the data set has an odd number of values, the median is the exact middle value. If the data set has an even number of data, then the two middle numbers are averaged.
Find the median of the following data set: {75, 62, 78, 92, 83, 91}
Place the data in ascending order: {62, 75, 78, 83, 91, 92}
median = \(\frac{78\ +\ 83}{2}\)
= 80.5
Adding a constant to each value in a data set will change both the mean and median by the same constant. Multiplying or dividing each value in a set by a constant will scale the mean and median by the same amount.
Outliers are values in a data set that are much larger or smaller than the other values in the set. Outliers can skew the mean, making it higher or lower than most of the values. The median is not affected by outliers, so it is a better measure of central tendency when outliers are present.
For the set {3, 5, 10, 12, 65}:
mean = \(\frac{3\ +\ 5\ +\ 10\ +\ 12\ +\ 65}{5}\)
= 19
median = 10
The mode is simply the value that occurs most often. There can be one, several, or no modes in a data set.
Find the mode: {16, 19, 19, 25, 27, 29, 75}
mode = 19