Practice Maths

L51 — Median, Mode and Range

Median, Mode and Range

Median = the middle value when data is ordered from smallest to largest.

  • Odd count: the middle value is the median
  • Even count: average the two middle values

Mode = the value that appears most often. A dataset can have one mode, multiple modes, or no mode.

Range = maximum − minimum. Measures how spread out the data is.

Hot Tip: Data MUST be ordered smallest to largest before finding the median. Write the ordered list first, then count to find the middle.

Worked Example

Odd count: {3, 7, 2, 7, 9, 4, 5}

Ordered: 2, 3, 4, 5, 7, 7, 9    (7 values → middle is position 4)

Median = 5  |  Mode = 7  |  Range = 9 − 2 = 7

Even count: {2, 5, 8, 11}

Ordered: 2, 5, 8, 11    (two middle values)

Median = (5 + 8) ÷ 2 = 6.5  |  No mode  |  Range = 11 − 2 = 9

Key Terms

median
the middle value of an ordered dataset; half the values are below it and half are above
mode
the value that appears most often in a dataset
range
a measure of spread: range = maximum − minimum
bimodal
a dataset with two modes (two values that each appear more often than any other)
measures of centre
values that describe the typical or middle value of a dataset — mean, median, and mode are all measures of centre

The Median

The median is the middle value once data is sorted from smallest to largest. It splits the data in half — 50% of values sit below it, 50% above.

  • Odd number of values: the median is the single middle value. For 7 values, it is in position 4.
  • Even number of values: average the two middle values. For 8 values, average positions 4 and 5.
  • Critical rule: always sort the data from smallest to largest before finding the middle value (median).

The Mode

The mode is the most frequently occurring value. Key points:

  • A dataset can have one mode, two modes (bimodal), or no mode (all values appear equally often).
  • The mode is most useful for categorical data — e.g. which shoe size sells most, which colour is most popular.

The Range

Range = maximum − minimum. It measures spread, not centre. A large range means values are widely spread; a small range means they cluster together. Because it only uses two values, a single extreme value can greatly affect it.

When to Use Each Measure

  • Mean — use when data is balanced with no outliers; it uses every value.
  • Median — use when data has outliers or is skewed (e.g. house prices, salaries); it ignores extremes.
  • Mode — use for categories or when the most common value is what matters.
  • Range — always report alongside a measure of centre to give a complete picture.
Memory aid — SORT FIRST: Write "SORT" at the top of any question asking for the median. Forgetting to sort is the most common error in this topic. Once sorted, circle the middle value (or two middle values) to make your working clear.

  1. Find Median, Mode and Range

    For each dataset, find the median, mode and range. Order the data first.

    1. {4, 8, 3, 6, 5}
    2. {12, 7, 12, 9, 8, 12, 5}
    3. {3, 3, 5, 7, 9, 11, 11}
    4. {2, 4, 6, 8, 10}
    5. {15, 8, 22, 8, 17, 8}
    6. {1, 2, 3, 4, 5, 6, 7, 8, 9}
    7. {30, 25, 30, 40, 35, 25}
    8. {7, 9, 11, 13, 15, 17}

  2. Which Measure to Use?

    1. A shoe shop wants to know which shoe size to stock the most of. Should they use mean, median, or mode? Explain why.
    2. Salaries at a company are: $42,000; $45,000; $48,000; $50,000; $52,000; $280,000. Should the company report the mean or median as the "typical" salary? Why?
    3. For {3, 5, 5, 6, 7, 8, 9}, the mean is approximately 6.1 and the median is 6. Are these measures similar? What does this suggest about the dataset?
    4. Describe a situation where the mean and median would give very different results.

  3. Outliers and Their Effect

    1. Dataset: {8, 9, 10, 10, 11, 12, 45}. Identify the outlier. Calculate the mean with and without it.
    2. Using the same dataset, calculate the median with and without the outlier. Is the median more or less affected than the mean?
    3. Explain in your own words why the median is called a "resistant" measure of centre.
    4. Test results: 55, 60, 62, 65, 68, 70, 72, 98. A teacher says the class "averaged" 69. Calculate both the mean and median. Which is more representative of a typical student's result?

  4. Median & Range in Context

    1. House prices — Street A: $420k, $450k, $480k, $500k, $530k. Street B: $310k, $390k, $480k, $590k, $720k. Find the median and range for each street. Which is more affordable? Which is more variable in price?
    2. Shoe sizes in a class: 6, 7, 7, 8, 8, 8, 9, 9, 10, 6, 7, 8, 9, 8, 7. Find the mode, median, and range.
    3. Two batters' scores across 5 innings — Batter A: 12, 45, 8, 55, 30. Batter B: 28, 31, 35, 29, 32. Find the range and median for each. Which batter is more consistent? Which has the higher peak?
    4. A dataset of 7 values has a median of 12. Six of the values are: 5, 9, 11, 13, 16, 20. The seventh value falls between 11 and 13. Find it.

  5. Calculate from a Data Table

    The table shows daily temperatures (°C) recorded over one week.

    DayMonTueWedThuFriSatSun
    Temp (°C)22192519283124
    1. Order the temperatures from lowest to highest.
    2. Find the median temperature.
    3. Find the mode temperature.
    4. Find the range of temperatures.
    5. On which day was the median temperature recorded?

  6. True or False?

    State whether each statement is True or False and explain your reasoning.

    1. A dataset can have more than one mode.
    2. The median of {3, 5, 7, 9} is 7.
    3. If all values in a dataset are the same, the range is 0.
    4. Adding a very large value to a dataset will always change the median significantly.
    5. A dataset with no repeated values has no mode.
    6. The range tells you about the spread but not the centre of the data.

  7. Spot the Error

    A student was asked to find the median of {9, 3, 7, 5, 1}. Their working is shown below.

    Student's working:

    Dataset: 9, 3, 7, 5, 1

    Middle value (3rd): 7

    Median = 7

    1. What critical step did the student skip?
    2. Show the correct working and give the correct median.
    3. Why must data always be ordered before finding the median?

  8. Compare Two Teams

    Two football teams recorded goals scored in their last 7 matches.

    TeamGoals per match
    Sharks2314253
    Eagles0610725
    1. Find the median and range for each team.
    2. Find the mode for each team.
    3. Which team is more consistent? Use the range to justify your answer.
    4. Which team would you rather watch? Use statistics to support your opinion.

  9. Which is Correct?

    Two students each found the median of {4, 9, 2, 7, 5, 8}.

    Student A

    Ordered: 2, 4, 5, 7, 8, 9

    Two middle values: 5 and 7

    Median = (5 + 7) ÷ 2 = 6

    Student B

    Ordered: 2, 4, 5, 7, 8, 9

    6 values → middle is 3rd value = 5

    Median = 5

    1. Which student is correct? Explain.
    2. What mistake did the other student make?
    3. Explain in your own words how to find the median when there is an even number of values.

  10. Extended Investigation

    A physiotherapist records patients' pain scores (out of 10) before and after treatment.

    PatientBeforeAfter
    184
    263
    395
    474
    552
    683
    1. Find the median pain score before treatment.
    2. Find the median pain score after treatment.
    3. Find the range for before and after treatment.
    4. Write two sentences describing what the statistics suggest about the effectiveness of the treatment.
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