statistics basics

Mean, Median, and Mode

Pre-Algebra20 minDifficulty

You should know: variables and expressions, fractions and decimals

Overview

The mean, median, and mode are measures of central tendency -- numbers that summarize where data tends to cluster. The mean (average) is the sum divided by count. The median is the middle value when sorted. The mode is the most frequent value. Each captures a different aspect of 'typical.'

Intuition

Mean = fair share: if you split total equally. Median = middle person: line everyone up, pick the center. Mode = most popular: which value appears most often. If a billionaire joins a room of average-income people, the mean income jumps dramatically but the median barely changes -- the median is more robust to outliers.

Formal Definition

Definition

For a dataset {x_1, ..., x_n}: Mean = (x_1 + ... + x_n)/n. Median: sort the data; if n is odd, median is the middle value x_{(n+1)/2}; if n is even, median is the average of x_{n/2} and x_{n/2+1}. Mode: the value(s) occurring with highest frequency (a dataset may have no mode, one mode, or several modes).

\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i

Arithmetic mean

\text{Median} = \begin{cases} x_{(n+1)/2} & n \text{ odd} \\ \frac{x_{n/2} + x_{n/2+1}}{2} & n \text{ even} \end{cases}

Median

Notation

Notation	Meaning
$\bar{x}$	Sample mean
$\text{Md}$	Median
$\text{Mo}$	Mode

Theorems

Theorem 1: Mean vs Median and Skewness

For symmetric distributions, mean = median. For right-skewed (long right tail) data, mean > median. For left-skewed data, mean < median. The median is resistant to outliers; the mean is not.

Worked Examples

1
Sort: 3, 3, 3, 5, 7, 8, 9 (7 values).
2
Mean: sum = 38, count = 7.
$\bar{x} = 38/7 \approx 5.43$
3
Median: middle (4th) value is 5.
$\text{Md} = 5$
4
Mode: 3 appears 3 times (most frequent).
$\text{Mo} = 3$

✓ Answer

Mean ≈ 5.43, Median = 5, Mode = 3.

Practice Problems

Easyfill in blank

Find the mean of: 4, 8, 6, 12, 10.

Common Mistakes

Common Mistake

Forgetting to sort the data before finding the median.

Always sort first. The median is the middle of the sorted list, not the middle of the original order.

Quiz

Which measure is most resistant to extreme outliers?

MeanMedianModeSum

Historical Background

The arithmetic mean has been used since antiquity for practical purposes. Francis Galton popularized the median in the late 19th century as a robust alternative to the mean. The concept of mode was formalized in the early 20th century. These three measures became fundamental to descriptive statistics.

1880s
Francis Galton develops and promotes the median as a robust central measure
Francis Galton

Summary

Mean = sum / count. Sensitive to outliers.
Median = middle value of sorted data. Robust to outliers.
Mode = most frequent value. Can be none, one, or several.
For symmetric data, mean = median = mode approximately.

References

BookLarson, R. Pre-Algebra. Holt McDougal, 2010.