Original learning companion, not full textbook exercise solutions. Covers direct-method grouped mean and standard equal-width grouped mode/median calculations. Assumed-mean and step-deviation methods, irregular-class cases and all textbook exercises remain outside this companion.
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LESSON 1
Estimate a grouped mean
Use class marks and frequencies, explaining the grouping assumption.
The arithmetic mean is total value divided by the number of observations.
Replace each class by a representative
For classes 0–10, 10–20 and 20–30, the class marks are 5, 15 and 25.
Attach the frequencies
Suppose frequencies are 2, 3 and 5. The total number of observations is 10.
Build the weighted total
Σfx = 2×5 + 3×15 + 5×25 = 10 + 45 + 125 = 180.
Divide and state the limit
Estimated mean = 180/10 = 18. Grouped data hide exact individual values, so the result uses the class-mark approximation.
Worked example: For class marks 10, 20, 30 with frequencies 1, 2, 1, weighted total is 10 + 40 + 30 = 80 and total frequency is 4. The estimated mean is 20.
Watch for this: Classes with different frequencies do not get equal weight. A grouped mean is generally an estimate of the raw-data mean.
Discuss: Create two grouped distributions with the same classes but different frequencies. Explain how moving frequency toward the high class changes the estimated mean.
LESSON 2
Estimate a grouped mode
Identify the modal class before substituting frequencies.
The modal class has the greatest frequency; its midpoint need not be the estimated mode.
Locate the highest frequency
Classes 0–10, 10–20, 20–30 have frequencies 3, 8, 5. Thus 10–20 is the modal class.
Assign symbols by position
l = 10, h = 10, f₁ = 8, f₀ = 3 and f₂ = 5.
Substitute carefully
Mode = 10 + [(8 − 3)/(16 − 3 − 5)] × 10 = 10 + 50/8.
Interpret the estimate
Estimated mode = 16.25, which lies in 10–20. This estimates a concentration within the class, not an exact known raw observation.
Worked example: For neighbouring frequencies 4, 12, 8 around modal class 20–30, l = 20 and h = 10. Mode = 20 + (8/12)×10 = 80/3, about 26.67.
Watch for this: f₀ and f₂ mean frequencies of the neighbouring classes, not their class marks. Irregular or tied peaks need careful interpretation.
Discuss: Hold the peak frequency fixed and increase the next-class frequency. Use the formula to explore how the mode shifts toward that next class.
LESSON 3
Find the median class by cumulative frequency
Use the halfway position and interpolation within a class.
The median divides ordered observations into lower and upper halves.
Build running totals
For classes 0–10, 10–20, 20–30 with frequencies 2, 6, 2, the cumulative frequencies are 2, 8, 10.
Locate the halfway position
N = 10, so N/2 = 5. The running total passes 5 in the 10–20 class.
Substitute the correct values
l = 10, cf = 2, f = 6 and h = 10. Median = 10 + [(5 − 2)/6] × 10.
Interpret the interpolation
Estimated median = 15. The formula locates the halfway position within the class using a grouped-data approximation.
Worked example: For 0–10, 10–20, 20–30 with frequencies 3, 5, 2, N/2 = 5. The median class is 10–20. Median = 10 + ((5−3)/5)×10 = 14.
Watch for this: Use the cumulative frequency before the median class, not the cumulative total at its end. Use class boundaries consistently.
Discuss: Make a frequency table and a cumulative-frequency column. Explain why identifying only the largest frequency would find a mode class, not a median class.
