Measures of Central Tendency: Mean, Median, and Mode

Measures of Central Tendency

Etymology of 'Mean'

The term mean has two distinct etymological origins. In statistics, 'mean' refers to the average value and comes from the Latin medianus ("in the middle"), through Old French meien and Middle English mene. This sense is related to the idea of a "middle value" or central point among a set of numbers. The other sense of 'mean' (unkind/rude) comes from Old English gemæne, meaning "common, shared, public," and later evolved to mean "stingy, ignoble, nasty." These two senses are homonyms with no direct relation.

• "Average" sense: Romance/Latin in origin

• "Rude" sense: Germanic in origin

Numerically Summarizing Data

Key Characteristics

When summarizing data numerically, three main characteristics are considered:

• Shape of the distribution

• Center (average)

• Spread (variability)

The center and spread are numerical summaries. The center is commonly called the average, and in statistics, three versions are frequently used:

• Mean

• Median

• Mode

Notation

Symbols and Definitions

• N: Population size

• n: Sample size (total number of data values)

• xi: The i-th observation in the data set

Example: For the data set (4, 1, 5, 3, 6, 1, 3), n = 7; x1 = 4, x2 = 1, ..., x7 = 3.

Summation Notation

The uppercase Greek letter Σ (sigma) is used as a summation sign:

$\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n$

Example: For x1 = 4, x2 = 1, ..., x7 = 3:

$\sum_{i=1}^{7} x_i = 4 + 1 + 5 + 3 + 6 + 1 + 3 = 23$

Mean

Definition and Types

• Arithmetic mean: Computed by adding all the values of the variable in the data set and dividing by the number of observations.

• Population mean (parameter): Denoted by $\mu$ (pronounced "mew"), calculated using all individuals in a population.

• Sample mean (statistic): Denoted by $\bar{x}$ (pronounced "x-bar"), calculated using sample data.

Formulas:

Population mean:

$\mu = \frac{x_1 + x_2 + \cdots + x_N}{N} = \frac{\sum_{i=1}^{N} x_i}{N}$

Sample mean:

$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n}$

Remark: Parameters (population) are denoted by Greek letters; statistics (sample) by Latin letters.

Example: Computing Mean

Given travel times (in minutes) for 7 employees: 23, 36, 23, 18, 5, 26, 43, 32, 25

• Compute the population mean using all data.

• Compute the sample mean for random samples of n = 5 and n = 4 employees.

Median

Definition

The median of a variable is the value that lies in the middle of the data when arranged in ascending order. It is denoted by M for the sample median. The median divides the data so that 50% of the observations are above and 50% below this value.

Steps to Find the Median

1. Arrange the data in ascending order.

2. Determine the number of observations, n.

3. Find the middle observation:

   • If n is odd: Median is the $\frac{n+1}{2}$-th observation.

   • If n is even: Median is the arithmetic mean of the $\frac{n}{2}$-th and $\frac{n}{2} + 1$-th observations.

Example: Computing Median

Given travel times (in minutes) for 7 employees: 23, 36, 23, 18, 5, 26, 43, 32, 25

• Compute the population median using all data.

• Compute the sample median for random samples of n = 5 and n = 4 employees.

Mean vs Median

Comparing Measures of Central Tendency

When data are skewed or contain outliers, the mean may not represent the typical value. The median is more resistant to extreme values.

Example: Cell Phone Call Lengths

Call	Length (min)
1	7
2	4
3	48
4	3
5	6

Mean = 7.25, Median = 3.5. Since only one call is much longer than the mean, the median better describes the typical call length.

Resistant Numerical Summary

A numerical summary is resistant if extreme values (very large or small) do not affect its value substantially. The median is resistant; the mean is not.

Mode

Definition

The mode of a variable is the most frequent observation in the data set.

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

• Bimodal: Two modes

• Multimodal: Three or more modes

• Mode is usually not reported for multimodal data as it is not representative of a typical value.

Example: State of Birth of U.S. Vice Presidents

State	Tally	Frequency
Massachusetts	2	2
Virginia	2	2
New Jersey	2	2
New York	5	5
Other States	1	1

The mode is New York (highest frequency).

Example: Exam Scores

Data: 82, 77, 90, 71, 62, 68, 74, 84, 94, 88. Each value occurs only once; there is no mode.

Distribution Type vs Measure of Central Tendency

Relationship Table

Distribution Shape	Mean vs Median vs Mode
Perfectly symmetric data set	Mean = Median = Mode
Rightward skewness (extremely high values)	Mean > Median > Mode
Leftward skewness (extremely low values)	Mean < Median < Mode

Graphical Representations

Normal Distribution

In a symmetric, bell-shaped distribution, mean and median coincide at the center.

Right Skewed Distribution

Mean is greater than median; both are greater than mode.

Left Skewed Distribution

Mean is less than median; both are less than mode.

Best Measure of Central Tendency

• If the data set is symmetric and has no outliers, use the mean.

• If the data set is skewed or contains outliers, use the median.

• If the data set is exactly symmetric, all measures coincide: $\text{mean} = \text{median} = \text{mode}$

Examples: Measures of Center

Birth Weights Example

Given birth weights of 50 babies, compute mean ($\bar{x} = 7.49$) and median ($M = 7.35$). The distribution is bell-shaped and symmetric, so the mean is the best measure of central tendency.

Comparing Data Sets

• Data set (A): 20, 130, 140, 150, 270. Mean = 142, Median = 140. Distribution is approximately symmetric; use mean.

• Data set (B): 2, 20, 130, 140, 150, 270, 1003. Mean = 245, Median = 140. Distribution is skewed right due to outliers; use median.

Remark: The median is less sensitive than the mean to extremely large or small measurements (outliers).