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Fundamentals of Business Mathematics and Statistics · Measures of Central Tendency and Dispersion

Skewness and Kurtosis: Formulas, Types and Examples

Updated 10 October 2026 · Fact-checked

Skewness measures how lopsided a distribution is. Kurtosis measures how peaked or flat it is. To solve questions, use Karl Pearson's coefficient (Mean − Mode) ÷ SD or Bowley's coefficient (Q3 + Q1 − 2×Median) ÷ (Q3 − Q1). A positive answer means a longer right tail; a negative answer means a longer left tail.

Understand Skewness and Kurtosis

Skewness tells you whether a distribution is symmetrical. In a symmetrical distribution, the left and right halves mirror each other. The mean, median and mode are all equal. The skewness is zero.

When the distribution is not symmetrical, it is skewed. In positive skewness, the longer tail is on the right. A few very large values pull the mean up. So Mean > Median > Mode. In negative skewness, the longer tail is on the left. A few very small values pull the mean down. So Mean < Median < Mode. Income data is a common example of positive skew, because a few very high earners lift the mean.

The sign of the coefficient gives the direction. The size gives the degree. Skewness is a pure number with no units, so you can compare two distributions even when they use different units. That is why the coefficients divide by SD or by the quartile range.

Kurtosis is about the peak, not the tilt. It tells you how flat or sharp the top of the curve is compared with a normal curve. A mesokurtic curve is normal, with a medium peak. A leptokurtic curve is more peaked, with a sharp top and heavy tails. A platykurtic curve is flatter, with a low broad top.

Remember the difference: skewness is about left-right balance, kurtosis is about peakedness. A curve can be symmetrical and still be peaked or flat.

Key formulas to remember

Karl Pearson's coefficient of skewness
Sk = (Mean − Mode) ÷ Standard Deviation
Use when the mode is well defined. Written as (X̄ − Z) ÷ σ.
Pearson's coefficient using median
Sk = 3 × (Mean − Median) ÷ Standard Deviation
Use when the mode is not given or is ill-defined. It comes from the empirical relation Mode = 3 Median − 2 Mean.
Bowley's coefficient of skewness
Sk = (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1)
Based on quartiles only. It always lies between −1 and +1.
Empirical relation
Mode = 3 × Median − 2 × Mean
Use to find a missing mode when mean and median are given.
Order of averages
Positive skew: Mean > Median > Mode. Negative skew: Mean < Median < Mode
Zero skew: all three are equal.
Kurtosis types
Mesokurtic = normal peak; Leptokurtic = more peaked; Platykurtic = flatter
Kurtosis describes peakedness, not direction.

How to solve Skewness and Kurtosis questions

Use this method for any numerical or theory question on skewness and kurtosis.

  1. 1Read what is given: mean, median, mode, SD, or quartiles. This decides the formula.
  2. 2If quartiles and median are given, use Bowley's formula. If mean, mode and SD are given, use Pearson's first formula.
  3. 3If the mode is missing, use Mode = 3 Median − 2 Mean, or use Sk = 3 (Mean − Median) ÷ SD directly.
  4. 4Substitute carefully. Keep the order: Mean minus Mode, and Q3 + Q1 minus twice the median.
  5. 5Calculate the value and keep its sign.
  6. 6Read the sign: positive means right-skewed, negative means left-skewed, zero means symmetrical.
  7. 7For kurtosis questions, match the description (sharp, normal, flat) to leptokurtic, mesokurtic or platykurtic.

Quickest way: Sign first, then calculate

When to use it: Use this when the options differ in sign or when the question only asks about the nature of skewness.

  1. Compare mean and median, or mean and mode. If mean is larger, the skew is positive. If smaller, negative.
  2. Eliminate any option with the wrong sign before calculating.
  3. For Bowley, check the numerator first: Q3 − Median against Median − Q1. The bigger gap shows the direction.
  4. Use simple numbers and divide at the end.
  5. Remember Bowley's value cannot go beyond −1 to +1. Reject options outside this range.

Common mistakes in Skewness and Kurtosis

  • Writing Mode − Mean instead of Mean − Mode in Pearson's formula.

    Students reverse the order and the sign flips.

    Fix: Always start with Mean. Positive skew means Mean is bigger, so the answer must be positive.

  • Forgetting the 2 in Bowley's numerator, writing Q3 + Q1 − Median.

    The formula is memorised loosely.

    Fix: Write it as [(Q3 − Median) − (Median − Q1)] ÷ (Q3 − Q1). This shows why Median is subtracted twice.

  • Confusing skewness with kurtosis.

    Both describe the shape of a curve.

    Fix: Skewness is tilt (left or right). Kurtosis is peak height (sharp or flat).

  • Mixing up leptokurtic and platykurtic.

    The names sound alike.

    Fix: Leptokurtic means more peaked, like a leap upward. Platykurtic means flat, like a plateau.

  • Calling a negative coefficient 'no skewness'.

    Students think negative means absent.

    Fix: Negative means left-skewed. Only zero means symmetrical.

  • Using the wrong SD, such as variance, in the denominator.

    Variance and SD are both given in the question.

    Fix: Divide by SD, not variance. If variance is given, take its square root first.

Worked examples

Example 1

For a distribution, Mean = 52, Mode = 46 and Standard Deviation = 12. Find Karl Pearson's coefficient of skewness and state the nature of skewness.

Show the solution
  1. Sk = (Mean − Mode) ÷ SD
  2. Sk = (52 − 46) ÷ 12
  3. Sk = 6 ÷ 12 = 0.5
  4. The value is positive, so the distribution is positively skewed.

Answer: Sk = +0.5; the distribution is positively skewed (longer right tail).

Example 2

The quartiles of a distribution are Q1 = 20, Median = 30 and Q3 = 50. Find Bowley's coefficient of skewness and state the nature of skewness.

Show the solution
  1. Sk = (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1)
  2. Numerator = 50 + 20 − 2 × 30 = 70 − 60 = 10
  3. Denominator = 50 − 20 = 30
  4. Sk = 10 ÷ 30 = 0.33 (approximately)
  5. Check: Q3 − Median = 20 and Median − Q1 = 10. The upper gap is larger, so the skew is positive.

Answer: Sk ≈ +0.33; the distribution is moderately positively skewed.

Exam tips

  • Most questions are direct substitution. Learn both Pearson formulas and Bowley's formula cold.
  • Check the sign first. It often eliminates two options at once.
  • Questions often ask for the order of mean, median and mode. Learn: positive skew Mean > Median > Mode; negative skew is the reverse.
  • Know the three kurtosis names and what each means. These come as one-line theory MCQs.
  • When mode is missing, use Mode = 3 Median − 2 Mean before applying Pearson's first formula.

Practice questions from Measures of Central Tendency and Dispersion

Skewness and Kurtosis in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Skewness and Kurtosis: frequently asked questions

What is the difference between skewness and kurtosis?

Skewness measures the lack of symmetry, that is, whether the tail is longer on the left or right. Kurtosis measures how peaked or flat the curve is compared with a normal curve. One is about tilt, the other about peak.

What is the Karl Pearson coefficient of skewness formula?

It is (Mean − Mode) ÷ Standard Deviation. If the mode is not available, use 3 × (Mean − Median) ÷ Standard Deviation. The sign shows the direction of skewness.

How are mean, median and mode related in positive and negative skewness?

In positive skewness, Mean > Median > Mode. In negative skewness, Mean < Median < Mode. In a symmetrical distribution, all three are equal.

What range does Bowley's coefficient take?

Bowley's coefficient lies between −1 and +1. A value of zero means the distribution is symmetrical in terms of quartiles. Values near ±1 show strong skewness.