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Lesson 4 of 4

Statistics · Lesson 4 of 4

Chapter Summary and Practice

“Mean, mode and median gather for a final tour through grouped data.”

Learning Objectives

• Recall all formulas for grouped mean, mode and median. • Choose an efficient calculation method. • Distinguish modal class from median class. • Compare the meaning of mean, median and mode. • Solve mixed grouped-data problems.

Mean, median and mode all describe the centre of a distribution, but in different ways. Mean is the arithmetic balance point, median is the middle position, and mode is the most frequent value.

Mean

Direct methodLaTeX
Assumed meanLaTeX
Step-deviationLaTeX

Mode

ModeLaTeX

Median

MedianLaTeX

Relationship Among the Three Measures

Empirical relationshipLaTeX
Important

This relationship is empirical. It is useful as an estimate, but it is not an exact identity for every possible distribution.

Which Measure Should You Use?

MeasureBest interpreted asUseful when
MeanOverall arithmetic averageAll values should contribute
MedianMiddle valueExtreme values may distort the mean
ModeMost frequent valueMost common or most popular value matters

How to Choose the Right Method

ClueMethod
Small class marksDirect mean
Large class marksAssumed mean
Convenient common factor in deviationsStep-deviation
Largest frequencyMode
Middle position / cumulative frequencyMedian
Known total plus missing frequencyForm an equation using the relevant formula
Common Mistakes

• Using class limits instead of class marks for mean, • Dividing by number of classes instead of total frequency, • Reporting the modal class as the mode, • Using the wrong cf in the median formula, • Confusing ordinary frequency with cumulative frequency, • Applying mode or median formulas before making intervals continuous where required,

Guided Practice

Guided Example 1: Mean

Problem
Intervals 500–520, 520–540, 540–560, 560–580, 580–600 have frequencies 12, 14, 8, 6, 10. Find the mean.

  1. 1.Class marks: 510, 530, 550, 570, 590.
  2. 2.Take a = 550, h = 20.
  3. 3.uᵢ = −2, −1, 0, 1, 2.
  4. 4.fᵢuᵢ = −24, −14, 0, 6, 20, so Σfᵢuᵢ = −12.
  5. 5.Σfᵢ = 50.
  6. 6.x̄ = 550 + 20(−12/50) = 545.2.
Guided Example 2: Mode

Problem
Intervals 0–20, 20–40, 40–60, 60–80, 80–100, 100–120 have frequencies 10, 35, 52, 61, 38, 29. Find the mode.

  1. 1.Modal class = 60–80.
  2. 2.l = 60, h = 20, f₁ = 61, f₀ = 52, f₂ = 38.
  3. 3.Mode = 60 + [(61 − 52)/(122 − 52 − 38)] × 20.
  4. 4.Mode = 65.625.
Guided Example 3: Median

Problem
Weights 40–45, 45–50, 50–55, 55–60, 60–65, 65–70, 70–75 have frequencies 2, 3, 8, 6, 6, 3, 2. Find the median.

  1. 1.Cumulative frequencies: 2, 5, 13, 19, 25, 28, 30.
  2. 2.n = 30, so n/2 = 15.
  3. 3.Median class = 55–60.
  4. 4.l = 55, cf = 13, f = 6, h = 5.
  5. 5.Median = 55 + [(15 − 13)/6] × 5 ≈ 56.67.
Guided Example 4: Compare All Three

Problem
Electricity-use intervals 65–85, 85–105, 105–125, 125–145, 145–165, 165–185, 185–205 have frequencies 4, 5, 13, 20, 14, 8, 4. Compare mean, median and mode.

  1. 1.Using class marks gives mean ≈ 137.06.
  2. 2.Cumulative frequencies are 4, 9, 22, 42, 56, 64, 68, so median class = 125–145 and median = 137.
  3. 3.Modal class = 125–145, giving mode ≈ 135.77.
  4. 4.The three values are close, but each describes a different feature of the distribution.
Guided Example 5: Empirical Relationship

Problem
Mean = 42 and median = 45. Estimate the mode.

  1. 1.Use 3 Median = Mode + 2 Mean.
  2. 2.135 = Mode + 84.
  3. 3.Mode = 51.

Quiz

Quick check

Which value represents a class interval when finding grouped mean?

Quick check

The modal class is the class with:

Quick check

To locate the median class, compare cumulative frequencies with:

Quick check

In the median formula, cf is:

Quick check

Which measure describes the most common value?

Before You Finish the Chapter

Check that you can calculate class marks, construct cumulative frequencies, identify modal and median classes, choose the correct formula, and explain what each final answer means.

Practice Problems

Practice Questions
  1. Find the mean for intervals 45–55, 55–65, 65–75, 75–85, 85–95 with frequencies 3, 10, 11, 8, 3.
  2. Find the mode for age intervals 5–15, 15–25, 25–35, 35–45, 45–55, 55–65 with frequencies 6, 11, 21, 23, 14, 5.
  3. Find the median for electricity-use intervals 65–85, 85–105, 105–125, 125–145, 145–165, 165–185, 185–205 with frequencies 4, 5, 13, 20, 14, 8, 4.
  4. If median = 28.5 for classes 0–10, 10–20, 20–30, 30–40, 40–50, 50–60 with frequencies 5, x, 20, 15, y, 5 and total 60, find x and y.
  5. Find mean, median and mode for surname-length intervals 1–4, 4–7, 7–10, 10–13, 13–16, 16–19 with frequencies 6, 30, 40, 16, 4, 4.
  6. If mean = 35 and median = 38, estimate the mode.
  7. If mean = 52 and mode = 46, estimate the median.
  8. Which measure would you choose for the most popular shoe size sold by a store, and why?
  9. Which measure would you choose for a typical household income when a few incomes are extremely high, and why?

Key Takeaways

Key Takeaways

• Mean uses all classes and their class marks. • Mode estimates the most frequent value. • Median locates the middle of the distribution using cumulative frequency. • Good table construction makes most grouped-data questions much easier.

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Median of Grouped Data

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