Question 1 (a)

Empirical relationship of mean, median and mode; when the median is the better average

3 marks

Time to write: About 6 minutes
Topic: Measures of central tendency

Question

Question 1 (a) · 3 marks

State the empirical relationship among mean, median and mode for symmetrical and asymmetrical distributions. In which situations is median regarded as a better average than the Arithmetic Mean?

Reworded from the university paper; the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

6 minfor 3 marks

Read from the question

State…Which…?2 asks, each with its own marks below. Answer both.
which situations is median regarded as…End with a clear choice and the reason for it.

Write it in this order

1 Relationship in symmetrical and asymmetrical distributions 1½ marks 3 min
2 Situations where the median is the better average 1½ marks 3 min

Short on time? Trim steps 1 and 2 evenly rather than skip one: each is capped at its own marks.

Try it first. About 6 minutes for 3 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Symmetrical distribution. In a symmetrical, single-peaked distribution the values are spread evenly on both sides of the centre, so the three averages coincide: Mean = Median = Mode. The normal curve is the standard example.

Asymmetrical (moderately skewed) distribution. The three separate, and the median lies between the mean and the mode. Karl Pearson's empirical rule links them: the distance between the mean and the mode is about three times the distance between the mean and the median,

Mean − Mode = 3(Mean − Median), or Mode = 3 Median − 2 Mean.

Positive skew (longer tail to the right): Mean > Median > Mode.
Negative skew (longer tail to the left): Mean < Median < Mode.

The rule holds only approximately, and only for moderately skewed distributions; it is used to estimate the mode when it is ill-defined.

When the median is better than the arithmetic mean.

Extreme values: incomes, wages, house prices or company sizes, where a few very large items would pull the mean up; the median is unaffected by how extreme they are.
Open-end classes: a distribution with below ₹10,000 or ₹1 lakh and above has no mid-point for the end classes, so the mean cannot be found, but the median can.
Qualitative or ranked data: intelligence, honesty or quality grades can be arranged in order but not measured; the middle-ranked item is the median.
Skewed distributions in general, where the typical (middle) case is wanted rather than a total-based average.
Incomplete data, where the number of items and the middle values are known but the extreme items are not.

Where the 3 marks sit

#	Component	Marks
1	Relationship in symmetrical and asymmetrical distributions	1.5
2	Situations where the median is the better average	1.5
	Total	3

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