Learning goals

What you should understand

  • Calculate mean, median, mode, and range from a small data set.
  • Choose a measure of center that matches the data type and distribution shape.
  • Explain how an extreme observation affects mean and median differently.
  • Recognize no-mode, multimodal, even-count, and categorical cases.
Before you begin

You should be able to order numbers, add signed values, divide, count frequencies, and distinguish quantitative values from category labels.

Three meanings of center and one endpoint spread

The mean uses every magnitude, the median uses ordered position, and the mode uses frequency. They can agree, but agreement is not required. Range answers a different question about endpoint spread.

Arithmetic meanxˉ=∑i=1nxin\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}
Medianx~=middle of the ordered data\widetilde{x}=\text{middle of the ordered data}
Modemode⁡=value or category with greatest frequency\operatorname{mode}=\text{value or category with greatest frequency}
RangeR=xmax⁡−xmin⁡R=x_{\max}-x_{\min}
  • Mean requires quantitative values for which addition and division are meaningful.
  • Median requires an ordered scale; for an even quantitative list, average the two middle values.
  • Mode can be used for quantitative or categorical data and can have no unique value or several tied values.

Reusable process

Choose the summary after examining the data

  1. 1
    Identify the data type

    Nominal categories support a mode but not a numerical mean or median. Quantitative observations can support all three.

  2. 2
    Order and count

    Sort values for the median and count frequencies for the mode.

  3. 3
    Calculate the candidate summaries

    Add and divide for mean, locate the middle for median, and compare frequencies for mode.

  4. 4
    Inspect shape and extremes

    An extreme magnitude can pull the mean while leaving the median near the central positions.

  5. 5
    Match the measure to the question

    Use mean for arithmetic balance, median for resistant ordered center, and mode for most frequent value or category.

Center comparison

Add an outlier and watch the centers respond

Compare 2, 3, 3, 4, 5 with the same list after adding 100. The control reveals the original and outlier cases without changing the definition of any measure.

2, 3, 3, 4, 5⟶2, 3, 3, 4, 5, 1002,\ 3,\ 3,\ 4,\ 5\quad\longrightarrow\quad2,\ 3,\ 3,\ 4,\ 5,\ 100
Data2, 3, 3, 4, 5
Mean3.4
Median3
Mode3

The mean, median, and mode describe the original five values.

Check the reasoning

Before the outlier, mean is 3.4, median is 3, and mode is 3. After adding 100, mean becomes 19.5, median becomes 3.5, and mode remains 3. The text fallback shows why the median and mode resist the outlier's magnitude more strongly than the mean.

Worked examples

Build from recognition to verification

Symmetric values without a repeated mode

The arithmetic and positional centers agree.

2, 4, 62,\ 4,\ 6
  1. Mean: The total is 12 across three values.
    xˉ=12/3=4\bar{x}=12/3=4
  2. Median: The ordered middle value is 4.
    x~=4\widetilde{x}=4
  3. Mode and range: No value repeats, and the endpoint span is 4.
    no mode,R=6−2=4\text{no mode},\quad R=6-2=4

Answer

xˉ=4,x~=4,no mode,R=4\bar{x}=4,\quad \widetilde{x}=4,\quad \text{no mode},\quad R=4

Check: The deviations from 4 are negative 2, 0, and 2, which balance to zero, and the ordered middle is visibly 4.

An extreme value separates mean and median

The largest observation changes the total much more than the middle position.

1, 2, 3, 4, 201,\ 2,\ 3,\ 4,\ 20
  1. Mean: The total is 30.
    xˉ=30/5=6\bar{x}=30/5=6
  2. Median: The third ordered value is 3.
    x~=3\widetilde{x}=3
  3. Interpret: The mean moves toward 20 while the median remains at the central position.
    6>36>3

Answer

xˉ=6,x~=3\bar{x}=6,\qquad \widetilde{x}=3

Check: Replacing 20 with 5 would reduce the mean to 3 while leaving the median at 3, showing that the extreme magnitude created the gap.

A bimodal numerical data set

Two values share the greatest frequency.

1, 1, 2, 2, 41,\ 1,\ 2,\ 2,\ 4
  1. Count frequencies: Both 1 and 2 appear twice.
    f(1)=f(2)=2f(1)=f(2)=2
  2. Find mean and median: The total is 10 and the third value is 2.
    xˉ=2,x~=2\bar{x}=2,\quad \widetilde{x}=2
  3. Report every mode: Neither tied peak should be discarded.
    modes⁡=1,2\operatorname{modes}=1,2

Answer

xˉ=2,x~=2,modes⁡=1,2\bar{x}=2,\quad \widetilde{x}=2,\quad \operatorname{modes}=1,2

Check: A direct frequency table confirms that 1 and 2 each occur more often than 4.

Categorical data use the mode

Color labels have frequencies but no arithmetic or numerical order.

blue, red, blue, green, blue\text{blue, red, blue, green, blue}
  1. Count categories: Blue appears three times; red and green appear once each.
    f(blue)=3f(\text{blue})=3
  2. Choose an appropriate summary: Only the mode is defined for nominal colors.
    mode⁡=blue\operatorname{mode}=\text{blue}
  3. Reject invalid arithmetic: Adding or ordering the category labels has no defined numerical meaning.
    mean and median are not defined\text{mean and median are not defined}

Answer

mode⁡=blue\operatorname{mode}=\text{blue}

Check: The frequency count is independent of the order in which the five category labels were observed.

Try it yourself

Practice the decision before checking the answer

  1. Find mean, median, mode, and range for 2, 2, 5, 7, 9.

    Show answer and explanation

    Answer: \bar{x}=5,\ \widetilde{x}=5,\ \operatorname{mode}=2,\ R=7

    The values total 25, the third value is 5, and 2 is the only repeat.

  2. Which center is usually more resistant for 20, 21, 22, 23, 400?

    Show answer and explanation

    Answer: The median, 22.

    The extreme value changes the total but not the middle ordered position.

  3. Can shirt size categories small, medium, and large have a median?

    Show answer and explanation

    Answer: They can have an ordered median, but an arithmetic mean is not meaningful without a justified numerical scale.

    Ordinal categories have rank order but unequal or undefined numerical spacing.

  4. What should be reported when two values tie for the greatest frequency?

    Show answer and explanation

    Answer: Report both modes.

    The mode definition does not require a unique value.

Common mistakes

Notice the decision before the algebra

Calling every average a mean

Using average without naming which center was calculated.

Fix: State arithmetic mean, median, or mode so the definition is clear.

Taking the typed middle as the median

Reading input position before sorting.

Fix: Order the values first, then use the central position or pair.

Forcing one mode

Choosing a single winner when several values tie or every value has equal frequency.

Fix: State the mode convention and report every tied nontrivial peak.

Ignoring data type and shape

Using the mean for categories or treating it as resistant to extreme values.

Fix: Choose a measure whose arithmetic, order, and sensitivity match the data and question.