8 Measures of Skewness
Central tendency locates the middle of a dataset, and dispersion measures how spread out it is, but neither one says anything about shape. Two datasets can share the same mean and the same standard deviation and still look completely different, one piled up on the left with a long tail stretching right, the other piled up on the right with a long tail stretching left. Skewness is the measure that captures this lopsidedness. This chapter covers Karl Pearson’s coefficients of skewness, Bowley’s quartile-based coefficient, and the moment-based coefficient, for both the raw data and the grouped frequency table carried forward from earlier chapters.
8.1 What Is Skewness?
A distribution is symmetric when its left and right halves are mirror images of each other, in which case the mean, median, and mode all coincide at the center. Real data is rarely perfectly symmetric.
- Positive (right) skew: A long tail stretches toward higher values. Most observations cluster at the lower end, with a few large values pulling the mean upward. Typically \(\text{Mean} > \text{Median} > \text{Mode}\). Household income and property prices are classic examples.
- Negative (left) skew: A long tail stretches toward lower values. Most observations cluster at the higher end, with a few small values pulling the mean downward. Typically \(\text{Mean} < \text{Median} < \text{Mode}\). Retirement age or exam scores on an easy test often look like this.
- Zero skew: The distribution is symmetric, and mean, median, and mode coincide.
A quick sanity check that needs no formula at all: just compare the mean and the median. If they are close, the data is roughly symmetric. If the mean is noticeably higher, the data is positively skewed; if it is noticeably lower, the data is negatively skewed. Every coefficient in this chapter is really a more precise, standardised version of that same comparison.
8.2 Karl Pearson’s Coefficient of Skewness
Pearson’s coefficients compare the mean to the mode or the median, then standardise the result by dividing by the standard deviation so that datasets in different units become comparable.
Mode-based (used when the distribution has a single, well-defined mode):
\[Sk = \frac{\text{Mean} - \text{Mode}}{SD}\]
Median-based (used when the mode is ill-defined, absent, or the dataset has multiple modes; also called the second coefficient, and derived from the empirical relationship \(\text{Mean} - \text{Mode} \approx 3(\text{Mean} - \text{Median})\)):
\[Sk = \frac{3(\text{Mean} - \text{Median})}{SD}\]
Example
For the seven test marks 58, 62, 58, 74, 65, 58, 70: mean \(\approx 63.57\), mode \(= 58\), median \(= 62\), and the population standard deviation is \(5.95\) (from Chapter 6).
\[Sk_{\text{mode}} = \frac{63.57 - 58}{5.95} \approx 0.94 \qquad Sk_{\text{median}} = \frac{3(63.57 - 62)}{5.95} \approx 0.79\]
Both coefficients are positive, confirming what the raw numbers already hinted at: mean (63.57) \(>\) median (62) \(>\) mode (58), so this small dataset is positively skewed, dragged upward by the two higher marks 74 and 65.
8.3 Bowley’s Coefficient of Skewness
Pearson’s coefficients use the mean and are therefore still somewhat sensitive to extreme values. Bowley’s coefficient sidesteps this entirely by using only the quartiles, making it a robust, outlier-resistant measure of skewness, well-suited to skewed data with open-ended classes or extreme observations.
\[Sk_{\text{Bowley}} = \frac{Q_3 + Q_1 - 2 \times \text{Median}}{Q_3 - Q_1}\]
The result always falls between \(-1\) and \(+1\).
Example
For the same seven marks (sorted: 58, 58, 58, 62, 65, 70, 74), using the \(\frac{n+1}{4}\) positional method, \(Q_1 = 58\) and \(Q_3 = 70\), with median \(= 62\).
\[Sk_{\text{Bowley}} = \frac{70 + 58 - 2(62)}{70 - 58} = \frac{4}{12} \approx 0.33\]
Again positive, agreeing with both Pearson coefficients.
8.4 Moment-Based Coefficient of Skewness
The most statistically rigorous measure uses the third moment about the mean, \(m_3 = \frac{\sum (x_i - \bar{x})^3}{n}\), standardised by the cube of the standard deviation. Because the deviations are cubed rather than squared, the sign is preserved: positive skew stays positive, negative skew stays negative.
\[Sk = \frac{m_3}{SD^3}\]
Example
For the same seven marks, \(m_3 \approx 125.72\) and \(SD^3 \approx 210.51\), giving \(Sk \approx 0.60\). All four figures computed in this chapter so far (0.94, 0.79, 0.33, 0.60) point the same direction: this small dataset is mildly to moderately positively skewed. They don’t have to match numerically, each coefficient is defined differently, but agreement in sign is what matters for interpretation.
8.5 Skewness for Grouped Data
All three coefficients extend to grouped data using the class marks and the grouped mean, median, mode, and quartiles already computed in Chapters 5 and 6. Continuing with the same 30-student marks distribution:
| Class Interval | Class Mark (\(x\)) | Frequency (\(f\)) |
|---|---|---|
| 30–40 | 35 | 2 |
| 40–50 | 45 | 4 |
| 50–60 | 55 | 6 |
| 60–70 | 65 | 6 |
| 70–80 | 75 | 5 |
| 80–90 | 85 | 4 |
| 90–100 | 95 | 3 |
Grouped mean \(= 65.67\), grouped median \(= 65.0\), grouped mode \(= 60.0\), grouped \(SD = 17.11\), \(Q_1 = 52.5\), \(Q_3 = 79.0\) (all from Chapters 5 and 6). For grouped data the third moment uses the same class-mark substitution: \(m_3 = \dfrac{\sum f_i (x_i - \bar{x})^3}{N}\).
Example
\[Sk_{\text{mode}} = \frac{65.67-60.0}{17.11} \approx 0.33 \qquad Sk_{\text{median}} = \frac{3(65.67-65.0)}{17.11} \approx 0.12\]
\[Sk_{\text{Bowley}} = \frac{79.0+52.5-2(65.0)}{79.0-52.5} \approx 0.06 \qquad Sk_{\text{moment}} \approx 0.06\]
All four are small but positive, so the full 30-student distribution is only mildly positively skewed, far less skewed than the small seven-value example. This makes sense: with 30 marks spread fairly evenly across seven classes, no single class dominates the way the repeated value 58 did in the smaller dataset.
8.6 Interpreting Skewness
- \(Sk > 0\) (positive/right skew): Tail stretches right; a few unusually large values pull the mean above the median.
- \(Sk < 0\) (negative/left skew): Tail stretches left; a few unusually small values pull the mean below the median.
- \(Sk \approx 0\): Roughly symmetric; the mean is a trustworthy measure of the center.
- Pearson vs. Bowley vs. Moment-based: Pearson’s coefficients are quick and intuitive but need the mode or median; Bowley’s is the most robust to outliers since it ignores everything outside the middle 50%; the moment-based coefficient is the most mathematically rigorous and the one used internally by most statistical software.
Skewness tells you the direction of the asymmetry, not how it should be handled. Highly skewed data (income, wait times, transaction amounts) often benefits from reporting the median instead of the mean, or from a log transformation before further analysis, a technique this book returns to in a later module.
Looking Ahead
Skewness answers “which way does the distribution lean?” The next chapter answers a related but different question: “how peaked or flat is the distribution compared to a normal bell curve?” That measure is kurtosis, and together skewness and kurtosis complete the picture of a distribution’s shape, the last piece of descriptive statistics before this book moves on to probability and inferential statistics.
Summary
| Concept | Description |
|---|---|
| Core Idea | |
| Skewness | A measure of the asymmetry, or lopsidedness, of a distribution's shape |
| Symmetric Distribution | A distribution whose left and right halves mirror each other, where mean, median, and mode coincide |
| Positive Skew | A distribution with a long right tail; typically mean > median > mode |
| Negative Skew | A distribution with a long left tail; typically mean < median < mode |
| Coefficients | |
| Pearson's Coefficient (Mode-Based) | Mean minus mode, divided by the standard deviation |
| Pearson's Coefficient (Median-Based) | Three times mean minus median, divided by the standard deviation, used when the mode is unreliable |
| Bowley's Coefficient | Uses only the quartiles, Q3 plus Q1 minus twice the median, over Q3 minus Q1; robust to outliers |
| Moment-Based Coefficient | Third moment about the mean, divided by the cube of the standard deviation; the most rigorous measure |
| Grouped Data | |
| Grouped Skewness | All coefficients extend to grouped data using class marks and the grouped mean, median, mode, and quartiles |
| Interpretation | |
| Interpreting the Sign | A positive coefficient means right skew, negative means left skew, near zero means symmetric |
| Choosing a Coefficient | Pearson's coefficients are quick and intuitive, Bowley's is outlier-robust, the moment-based version is the statistical standard |