Three roads, one mean

The idea

All three methods give the same mean, the other two being simplified forms of the direct method; the direct method suits small xi and fi, the others large values, with step-deviation also for unequal class sizes.

Which method?

You now know three ways to find the mean of grouped data: direct, assumed mean and step-deviation.

Two NCERT examples: the percentage of female teachers in 35 states and union territories, and the wickets taken by 45 bowlers. In each we can use all three methods.

Will the three methods give three different means for the same table?

• Yes, three different answers
• No, the same answer every time
• Two will agree and one will not

Run all three methods

Tap Direct, Assumed and Step (step-deviation) for each of the two tables. Watch the means, and the biggest number you have to write in the last column.

Same mean, easiest road

All three methods give the same mean. The assumed mean and step-deviation methods are simplified forms of the direct method.

If the values of x and f are small, the direct method is a good choice. If they are large, use the assumed mean or the step-deviation method. If the class sizes are unequal and x is large, you can still use step-deviation by taking h to be a suitable divisor of all the deviations (in Example 3 the book takes h = 20).

Notes

All three methods give the same mean. Direct for small x and f; assumed mean or step-deviation for large values; step-deviation also for unequal class sizes.

Check yourself

The class marks are 2, 4, 6, 8, 10 and the frequencies are 7, 8, 2, 2, 1 (NCERT Example 5). All numbers are small. Which method is a good choice?

Example 3 has class marks 40, 80, 125, 200, 300, 400 and unequal class sizes. Which method keeps the biggest number in the last column smallest?

Example 3: a = 200, h = 20, Σfᵢuᵢ = −106 and Σfᵢ = 45. Find x̄, correct to 2 decimal places.

Answer: 152.89

20 × (−106/45) = −47.11…, and 200 − 47.11… = 152.88…, which is 152.89 to 2 decimal places.

Ravi finds a mean of 39.71 by the direct method and 41.2 by the assumed mean method on the same table. What does this tell you?

• Direct method — correct. Yes! The numbers are small, so f·x is easy to find and no extra column is needed.
• Step-deviation method. It works, but for such small numbers the extra columns d and u are more work than they save.
• None of them works. All three work on every table. The direct method is the easiest here.
• Direct: biggest f·x is 2400. The tool showed it: 12 × 200 = 2400.
• Assumed mean: biggest f·d is 1200. The tool showed it: 1200 for 16 × (−75).
• Step-deviation: biggest f·u is 60 — correct. Yes! With a = 200 and h = 20 the biggest product is 60 (16 × −3.75). This is the method the book uses.
• He made an arithmetic slip in one of them — correct. Yes! All methods must give the same mean, so the two answers cannot both be right.
• Different methods give different means. No. The assumed mean method is a simplified form of the direct method. The answers must agree.
• The assumed mean method is only approximate. It is not. It gives exactly the same value as the direct method.