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Quantitative Aptitude · Measures of Central Tendency and Dispersion

Skewness and Measures of Shape for CA Foundation

Updated 1 October 2026 · Fact-checked

Skewness measures how lopsided a distribution is. In positive skew the tail stretches right and mean > median > mode. In negative skew the order reverses. To solve, use Karl Pearson's (Mean − Mode) ÷ SD, or 3(Mean − Median) ÷ SD, or Bowley's quartile formula, then read the sign.

Understand Skewness and Measures of Shape

A distribution is symmetric when its left and right halves mirror each other. In a symmetric distribution the mean, median and mode are equal, and Q3 is as far above the median as Q1 is below it.

Skewness tells you how far a distribution departs from symmetry, and in which direction. If the long tail is on the right (a few very large values), the distribution is positively skewed. Income data is a common example. If the long tail is on the left, it is negatively skewed.

The mean is pulled toward the tail most, the mode least, and the median sits in between. So in positive skew: Mean > Median > Mode. In negative skew: Mean < Median < Mode. Remember that the mean is always the one nearest the tail.

The sign of the coefficient gives the direction. Positive means right-skewed, negative means left-skewed, zero means symmetric. The size tells you how strong the skew is. To make skewness comparable across data sets, the measures are divided by a measure of spread. This gives a pure number with no units.

Two coefficients are tested most. Karl Pearson's uses the mean, mode (or median) and standard deviation. Bowley's uses only the quartiles and median. It is handy for open-ended classes where the mean and SD cannot be found.

Key formulas to remember

Karl Pearson's coefficient (with mode)
Sk = (Mean − Mode) ÷ SD
Use when the mode is given or well defined. SD is the standard deviation, not the variance.
Karl Pearson's coefficient (with median)
Sk = 3 × (Mean − Median) ÷ SD
Use when the mode is ill-defined or not given. It follows from Mode = 3 Median − 2 Mean.
Empirical relation among averages
Mode = 3 Median − 2 Mean
Holds approximately for moderately skewed distributions. It is not an exact law.
Bowley's coefficient
Sk = (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1)
Always lies between −1 and +1. It depends only on the middle 50% of the data.
Order of averages
Positive skew: Mean > Median > Mode. Negative skew: Mean < Median < Mode. Symmetric: Mean = Median = Mode
Use this to check the sign of your answer.
Reading the sign
Sk > 0: positive skew. Sk < 0: negative skew. Sk = 0: symmetric
The sign is decided by the numerator in both coefficients.

How to solve Skewness and Measures of Shape questions

Follow this order for any skewness question. It keeps the sign and the choice of formula correct.

  1. 1Read what is given: mean, median, mode, SD, Q1, Q3. Also note what is asked: the coefficient, the direction, or a missing average.
  2. 2Pick the formula. If Q1, Q3 and the median are given, use Bowley's. If the mean, mode and SD are given, use (Mean − Mode) ÷ SD. If the mode is missing, use 3(Mean − Median) ÷ SD.
  3. 3If a missing average is needed, get it from Mode = 3 Median − 2 Mean, or rearrange the coefficient formula.
  4. 4Work out the numerator first, keeping its sign. Mean minus mode, or Q3 + Q1 minus twice the median.
  5. 5Divide by the SD, or by Q3 − Q1 for Bowley. Make sure you use SD and not variance.
  6. 6Read the sign: positive means right tail, negative means left tail, zero means symmetric.
  7. 7Check against the order of averages. A positive answer needs Mean > Median > Mode.

Quickest way: Sign first, then divide

When to use it: Use this in the MCQ paper when options differ in sign or size, so you can eliminate fast.

  1. Work out only the numerator and note its sign. This often removes two options at once.
  2. For Bowley, the answer must lie between −1 and +1. Reject any option outside that range.
  3. Compare the gap between mean and median. If Mean > Median, the answer is positive.
  4. Divide by SD last. Do simple fractions mentally, for example 6 ÷ 12 = 0.5.
  5. If the mode is missing, go straight to 3(Mean − Median) ÷ SD. Do not find the mode first.
  6. Skip the question if it needs long grouped-data calculation of mean and SD. Each wrong answer costs 0.25 marks.

Common mistakes in Skewness and Measures of Shape

  • Writing Mode − Mean instead of Mean − Mode

    Students reverse the order and the sign flips.

    Fix: Always write Mean first. Then check the sign against the order of averages.

  • Dividing by variance instead of SD

    The question gives variance, and students plug it in directly.

    Fix: Take the square root of the variance first. Skewness divides by SD.

  • Using Q3 − Q1 in the numerator of Bowley's formula

    Students mix the numerator with the denominator.

    Fix: Numerator is Q3 + Q1 − 2 × Median. Denominator is Q3 − Q1.

  • Calling a left-tailed distribution positive skew

    Students look at where the peak sits, not where the tail goes.

    Fix: Skew is named after the tail. A long right tail means positive skew.

  • Using the median-based formula without the factor 3

    Students write (Mean − Median) ÷ SD from memory.

    Fix: Remember 3 comes from Mode = 3 Median − 2 Mean. So Mean − Mode = 3(Mean − Median).

  • Accepting a Bowley value of 1.4 or −2

    Students do not check the range.

    Fix: Bowley's coefficient cannot go beyond ±1. If you get more, recheck your arithmetic.

Worked examples

Example 1

For a moderately skewed distribution, mean = 40, median = 38 and SD = 4. Using the empirical relation, Karl Pearson's coefficient of skewness is: (a) 0.5 (b) 1.5 (c) −1.5 (d) 3

Show the solution
  1. The mode is not given, so use Mode = 3 Median − 2 Mean.
  2. Mode = 3 × 38 − 2 × 40 = 114 − 80 = 34.
  3. Sk = (Mean − Mode) ÷ SD = (40 − 34) ÷ 4 = 6 ÷ 4 = 1.5.
  4. Check with the other form: 3 × (40 − 38) ÷ 4 = 6 ÷ 4 = 1.5. Same result.
  5. Mean > Median, so the sign is positive. This removes option (c).

Answer: (b) 1.5

Example 2

Karl Pearson's coefficient of skewness is −0.6, the mean is 52 and the SD is 10. The mode is: (a) 46 (b) 52 (c) 58 (d) 64

Show the solution
  1. Use Sk = (Mean − Mode) ÷ SD.
  2. Substitute: −0.6 = (52 − Mode) ÷ 10.
  3. Multiply both sides by 10: 52 − Mode = −6.
  4. So Mode = 52 + 6 = 58.
  5. Check: negative skew needs Mean < Mode. Here 52 < 58, which is correct.

Answer: (c) 58

Example 3

For a distribution Q1 = 20, median = 32 and Q3 = 40. Bowley's coefficient of skewness is: (a) 0.2 (b) −0.2 (c) 0.4 (d) −0.4

Show the solution
  1. Bowley's formula: (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1).
  2. Numerator = 40 + 20 − 2 × 32 = 60 − 64 = −4.
  3. Denominator = 40 − 20 = 20.
  4. Sk = −4 ÷ 20 = −0.2.
  5. The median is closer to Q3 than to Q1, so the left side is stretched. Negative skew fits.

Answer: (b) −0.2

Exam tips

  • Questions are mostly direct formula use. Learn both Pearson forms and Bowley's formula cold.
  • Check the sign first. The order of mean, median and mode often removes two options in seconds.
  • Read carefully whether SD or variance is given. Examiners use this as a trap.
  • Bowley's coefficient must lie within −1 and +1. Use this to catch arithmetic slips.
  • Expect conceptual questions on which average lies where for each skew type. Practise the order of averages until it is automatic.

Practice questions from Measures of Central Tendency and Dispersion

Skewness and Measures of Shape: frequently asked questions

What is the difference between positive and negative skewness?

In positive skewness the long tail is on the right and Mean > Median > Mode. In negative skewness the long tail is on the left and Mean < Median < Mode. The coefficient is positive in the first case and negative in the second.

What is Karl Pearson's coefficient of skewness formula?

It is (Mean − Mode) ÷ SD. When the mode is not available, use 3 × (Mean − Median) ÷ SD. The second form comes from the empirical relation Mode = 3 Median − 2 Mean.

When should I use Bowley's coefficient?

Use it when quartiles are given or when the data has open-ended classes, so the mean and SD cannot be found. It uses only Q1, Q3 and the median. Its value always lies between −1 and +1.

Is Mode = 3 Median − 2 Mean always true?

No. It is an empirical relation that holds approximately for moderately skewed distributions. In exam questions, use it when the question says moderately skewed or gives no mode.