Measures of Center in Statistics: Mean, Median, Mode, and Midrange

Part I: Basic Concepts of Measures of Center

Introduction to Measures of Center

Measures of center are statistical values that describe the central or typical value in a data set. They are essential for summarizing and interpreting data, providing a single value that represents the entire distribution. The four most common measures of center are the mean, median, mode, and midrange.

Mean

The mean, also known as the arithmetic mean, is what most people refer to as the average. It is the most widely used measure of center in statistics.

• Definition: The mean is calculated by adding all the data values and dividing by the total number of data values.

• Properties:

  • The mean is favored because it uses every value in its calculation.

  • It is sensitive to extreme values (outliers), which can skew the mean.

  • The mean is not always a value that appears in the data set.

Formula for the Mean:

If the data set is a sample, the mean is denoted by $\bar{x}$ (x-bar). If the data set is a population, the mean is denoted by $\mu$ (mu).

$\text{mean} = \frac{\text{The sum of all the data values}}{\text{The number of data values}}$

Or, using notation:

$\bar{x} = \frac{\sum x}{n}$

$\mu = \frac{\sum x}{N}$

• $\sum x$ = the sum of all the values

• $n$ = sample size (number of data values in a sample)

• $N$ = population size (number of data values in the population)

Example: For the data set 22, 22, 26, 24, 23:

$\bar{x} = \frac{22 + 22 + 26 + 24 + 23}{5} = \frac{117}{5} = 23.4$

Median

The median is the middle value when the data values are arranged in order. It divides the data set into two equal halves.

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

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

• The median is resistant to extreme values (outliers).

• The median does not use every data value in its computation.

Example:

• Data: 22, 22, 23, 24, 26 → Median = 23

• Data: 22, 22, 23, 24, 26, 27 → Median = (23 + 24)/2 = 23.5

Mode

The mode is the data value that occurs with the greatest frequency in a data set.

• A data set can have no mode, one mode, or more than one mode.

• If two data values occur with the same greatest frequency, the data set is bimodal.

• If more than two values occur with the same greatest frequency, the data set is multimodal.

• If no value is repeated, there is no mode.

• The mode is the only measure of center that can be used with categorical data.

Examples:

• Data: 22, 22, 24, 23 → Mode = 22

• Data: 22, 22, 23, 23, 24, 24, 26, 27 → Modes = 22, 23, 24 (multimodal)

• Data: 22, 23, 24, 26, 27 → No mode

Midrange

The midrange is the value midway between the maximum and minimum values in a data set. It is calculated as follows:

$\text{midrange} = \frac{\text{maximum data value} + \text{minimum data value}}{2}$

• The midrange is very sensitive to extreme values (outliers).

• It is rarely used but is easy to compute.

• It reinforces the idea that there are many different ways to find a center.

• Sometimes, the midrange is used when a median is desired but not available.

Example: For the data set 22, 22, 26, 24, 23:

$\text{midrange} = \frac{26 + 22}{2} = 24$

Common Notation

• $x$ = each individual data value

• $\sum x$ = the sum of all data values

• $n$ = sample size (number of data values in a sample)

• $N$ = population size (number of data values in the population)

• $\bar{x}$ = sample mean

• $\mu$ = population mean

Rounding

When calculating measures of center, round answers according to the context or as directed by your instructor. Typically, round to one more decimal place than the original data values.

Critical Thinking: Does it Make Sense?

It is important to consider whether it makes sense to calculate the mean or median for certain types of data. For example, it does not make sense to find the mean or median of:

• Different zip codes

• Rank of selected Amazon bestselling books

• Jersey numbers of the New England Patriots

In these cases, neither the mean nor the median is meaningful.

Comparison Table: Measures of Center

Measure	Definition	Usefulness	Sensitivity to Outliers
Mean	Sum of all values divided by the number of values	Most common, uses all data values	Very sensitive
Median	Middle value when data is ordered	Useful for skewed data	Not sensitive (resistant)
Mode	Most frequently occurring value	Useful for categorical data	Not sensitive
Midrange	Average of maximum and minimum values	Rarely used	Very sensitive

Additional info:

• When reporting measures of center, always consider the type of data and the presence of outliers.

• For skewed distributions, the median is often preferred over the mean.

• For categorical data, only the mode is appropriate.