Stem-and-leaf plots and grouped means

A stem-and-leaf plot keeps each observation

For scores 12, 14, 14, 18, 21 and 25, an ordered plot is

Key: 1 | 2 means 12 marks. Repeated leaves represent repeated observations. Without a key, the same digits could denote decimals or other units. A back-to-back plot compares two groups with a common stem scale.

A grouped mean needs representative values

For classes 0 < t ≤ 10, 10 < t ≤ 20, 20 < t ≤ 30, 30 < t ≤ 40 with frequencies 4, 8, 6, 2, use midpoints 5, 15, 25, 35. The estimated total is 4(5) + 8(15) + 6(25) + 2(35) = 360, so estimated mean is 360/20 = 18 minutes.

It is an estimate because every observation was replaced by its class midpoint. The actual total may differ. Do not average the four midpoints without frequency weights unless all class frequencies are equal.

Check your understanding 1 Weight the representative values

Two equal-width classes have midpoints 10 and 20 and frequencies 3 and 7. Estimate the mean. Explain why 15 is not the correct estimate.

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The estimated total is 3(10) + 7(20) = 170 across 10 observations, giving 17. The midpoint average 15 weights the classes equally even though seven of the ten observations are in the second class.