Practice Maths

L52 — Connecting Displays and Measures

Reading Measures from Data Displays

You can calculate all four measures (mean, median, mode, range) from a stem-and-leaf plot or dot plot by reading off all individual values first.

Steps:

  1. Read every value from the display carefully — it is easy to miss values
  2. List all values in order
  3. Calculate mean, median, mode, and range

Choosing the right measure:

  • Use mean when data is roughly symmetric with no outliers
  • Use median when data is skewed or has outliers
  • Use mode for categorical data or when the most common value matters
  • Use range to describe spread
Hot Tip: Read ALL values from the plot before calculating. In a stem-and-leaf plot, leaves like "0 0 6" mean three values (40, 40, 46), not two. Count the total number of leaves to confirm the count.

Worked Example

StemLeaves
12   4   7
20   3   3   8
31   5
42

Values: 12, 14, 17, 20, 23, 23, 28, 31, 35, 42    (10 values)

Mean = 245 ÷ 10 = 24.5  |  Median = (23+23)÷2 = 23  |  Mode = 23  |  Range = 42−12 = 30

Key Terms

symmetric
data that is balanced around the centre, with values spread roughly equally on both sides; when data is symmetric, mean ≈ median
skewed
data that is bunched to one side with a longer tail on the other; outliers cause skew and pull the mean away from the median
outlier
a value that is much higher or lower than the rest of the data; outliers strongly affect the mean but have little effect on the median
dot plot
a data display where each value is shown as a dot above a number line; the mode is the value with the most dots

Reading Values from a Display

Data displays contain all original values, ready for calculation. Before you can find any measure, list every value in order. Missing even one value gives wrong answers for everything that follows.

  • Stem-and-leaf plot: read each row left to right, combining stem with each leaf. The plot is already sorted, saving time for the median.
  • Dot plot: count the dots above each number — each dot is one value. The mode is immediately visible as the tallest column.

Calculating All Four Measures

Once you have a full sorted list:

  • Mean: add all values, divide by the count
  • Median: find the middle value (or average the two middle values for even count)
  • Mode: find the most frequently occurring value
  • Range: subtract the minimum from the maximum

Choosing the Best Measure

  • Mean — best for symmetric data with no outliers; uses every value
  • Median — best when data has outliers or is skewed (e.g. house prices, salaries); not affected by extremes
  • Mode — best for categories or when "most common" is what matters
  • Range — always report alongside a measure of centre to describe how variable the data is

Effect of Adding or Removing a Value

  • The mean always changes when any value is added or removed (it uses every value)
  • The median may shift slightly if the new value changes which values sit in the middle positions
  • The mode may change if a new value becomes the most frequent
  • Ask yourself: is the new value above or below the current mean? Does it change the middle positions?
Tip — reading carefully: In a stem-and-leaf plot, leaves like "3 3 5" have three values, not two — the 3 appears twice. Count all leaves to confirm the total count before calculating.

  1. Calculate All Four Measures

    For each dataset, find the mean, median, mode, and range. Show all working.

    1. 8, 12, 15, 9, 11, 12, 14, 10
    2. 3, 7, 3, 9, 5, 3, 8, 6, 4, 7
    3. 45, 52, 48, 61, 45, 58, 52, 49, 45, 60

  2. Measures from a Stem-and-Leaf Plot

    Ages of volunteers at a community event.

    StemLeaves
    16   8   9
    21   4   4   7
    32   5   8
    40   3
    51

    Key: 2 | 4 = 24 years old

    1. List all values in order.
    2. Calculate the mean, median, mode, and range.
    3. Write two sentences summarising what these measures tell you about the ages of volunteers.

  3. What Does Each Measure Tell You?

    1. A dataset of house prices has mean = $650,000 and median = $480,000. What does this difference suggest about the shape of the data? Which measure should a first-home buyer use when researching affordability?
    2. The range of a dataset is 0. What does this tell you about the values?
    3. For {5, 5, 5, 5, 100}, calculate all four measures. Explain which measure best describes the data and which is most misleading.

  4. Compare Two Classes

    Class 7C: 54, 61, 68, 72, 65, 70, 58, 74, 63, 71

    Class 7D: 42, 78, 65, 88, 50, 82, 59, 90, 46, 75

    1. Calculate the mean, median, and range for each class.
    2. Which class performed better overall, and which was more consistent? Refer to specific measures in your answer.
    3. A student in Class 7D says "Our class is better because we had the highest score." Is this a fair comparison? Explain using the measures you calculated.

  5. Complete the Measures Table

    Find the missing measures marked ?

    DatasetMeanMedianModeRange
    4, 6, 6, 8, 10??6?
    1, 3, 3, 5, 7, 9≈4.67???
    20, 20, 20, 20???0

  6. True or False?

    State True or False and give a reason.

    1. A dataset where mean = median is always perfectly symmetrical.
    2. The mode is the most useful measure for numeric data with no repeated values.
    3. The range can be calculated from a stem-and-leaf plot without listing all values.
    4. Two different datasets can have the same mean, median, and range.
    5. When mean > median, the data is likely skewed to the right (pulled up by high outliers).

  7. Spot the Error

    A student read the stem-and-leaf plot below and calculated four measures.

    StemLeaves
    23   5   8
    31   4
    40   0   6

    Student's working:

    Values: 23, 25, 28, 31, 34, 40, 46

    Mean = 227 ÷ 7 ≈ 32.4  |  Median = 31  |  Mode = none  |  Range = 23

    1. Identify the error in reading the plot.
    2. List the correct values and recalculate all four measures.

  8. Choose the Best Measure

    For each scenario, state which measure of centre is most appropriate and explain why.

    ScenarioBest MeasureReason
    Typical weekly wage in a country?Wages are skewed by very high earners; most people earn far less than the top end
    Most popular colour of car soldMode?
    Class test results with no outliers??
    Property prices in a suburb with one mansion??

  9. Reading from a Dot Plot

    The dot plot shows the number of books read by 12 students in one term.

    1 2 3 4 5 6 7

    Each dot = one student. Numbers on axis = books read.

    1. List all the data values shown.
    2. Find the mean, median, mode, and range.
    3. What does the mode tell you about students' reading habits?

  10. Extended Comparison

    A coach recorded laps completed by two training groups.

    GroupLaps completed
    Morning810911810129
    Evening514813715612
    1. Calculate mean, median, mode, and range for each group.
    2. Which group is more consistent? Which measure best supports this?
    3. Write a short report (3–4 sentences) comparing the two groups using the statistics you calculated.
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