What is the primary difference between a positively skewed and a negatively skewed distribution regarding their tails?

A positively skewed distribution has a long tail extending toward the right (higher values), while a negatively skewed distribution has a long tail extending toward the left (lower values). The 'skew' refers to the direction of this elongated tail.

How do the Mean and Median compare in a negatively skewed distribution?

In a negatively skewed distribution, the Mean is typically less than the Median. This happens because the extreme low values in the left tail pull the Mean downward more significantly than they affect the Median.

When should Bowley's Coefficient of Skewness be used instead of Pearson's Coefficients?

Bowley's Coefficient should be used when the dataset contains extreme outliers or is an open-ended distribution where the Mean and Standard Deviation cannot be accurately calculated. It relies on quartiles, which are more resistant to extreme values.

What is a common mistake when identifying skewness from a visual frequency polygon?

A common error is looking at the 'hump' or peak of the graph instead of the tail. Students often see the peak on the left and think 'left-skewed,' but if the peak is on the left, the tail is on the right, making it positively (right) skewed.

Why is it incorrect to assume that a skewness of zero always implies a perfectly symmetrical distribution?

While symmetry implies zero skewness, the reverse is not always true. In some complex or multimodal distributions, the asymmetries on different sides of the mean can mathematically cancel each other out, resulting in a skewness of zero despite the shape not being a mirror image.

What error occurs if you use the formula $(\text{Mode} - \text{Mean}) / \sigma$ for Pearson's coefficient?

The sign of the skewness will be reversed. The correct formula is $(\text{Mean} - \text{Mode}) / \sigma$, ensuring that if the Mean is larger (positive skew), the coefficient is positive.

Define Pearson's Second Coefficient of Skewness.

It is a measure of asymmetry calculated as $S_k = \frac{3(\text{Mean} - \text{Median})}{\sigma}$. It is often preferred over the first coefficient because the median is a more stable measure of central tendency than the mode.

What does a skewness value of $+1.5$ indicate about a dataset?

A value of $+1.5$ indicates a high degree of positive skewness. This means the distribution is significantly asymmetrical with a long tail on the right side and the mean is substantially higher than the median.

What is the formula for Bowley's Coefficient of Skewness?

The formula is $S_k = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$, where $Q_1$ is the first quartile, $Q_2$ is the median, and $Q_3$ is the third quartile.

In a perfectly symmetrical distribution, what is the relationship between the Mean, Median, and Mode?

In a perfectly symmetrical unimodal distribution, the Mean, Median, and Mode are all equal and located at the center of the distribution.

Skewness | AQA GCSE Statistics

Processing, Representing & Analysing Data

Skewness

Summary

Skewness is a fundamental statistical measure used to describe the degree of asymmetry in a probability distribution. It quantifies how much a distribution deviates from the symmetrical 'normal' bell curve, indicating whether data points are concentrated on one side and identifying the direction and relative length of the distribution's tails.

1. Definition & Core Concepts

Skewness refers to the lack of symmetry in a frequency distribution. In a perfectly symmetrical distribution, the left and right halves are mirror images of each other, and the central tendency measures (mean, median, and mode) coincide at the same point.
A distribution is said to be skewed when the data spreads out more significantly on one side of the center than the other. This creates a 'tail' that extends toward either the higher or lower values of the dataset.
The primary purpose of measuring skewness is to understand the shape of the data distribution beyond just its central location (mean) and spread (variance). It provides insight into the likelihood of extreme values occurring in one direction versus the other.

Comparison of Positive, Negative, and Symmetric distributions showing the direction of the tails.

2. Types of Skewness

Positive Skewness (Right-Skewed): Occurs when the tail on the right side of the distribution is longer or fatter than the left side. In this scenario, the majority of data points are clustered at the lower end of the scale, and the mean is typically greater than the median.
Negative Skewness (Left-Skewed): Occurs when the tail on the left side is longer or fatter. This indicates that the mass of the distribution is concentrated on the right, with a few extreme low values pulling the mean down below the median.
Zero Skewness: Represents a perfectly symmetrical distribution. While the Normal Distribution is the most famous example, any distribution where the left and right sides balance perfectly will result in a skewness value of zero.

3. Mathematical Measures of Skewness

4. Key Distinctions in Central Tendency

The relationship between the mean, median, and mode is the most reliable qualitative indicator of skewness. Because the mean is the most sensitive to extreme values, it is 'pulled' furthest toward the tail of the distribution.

Feature	Positive Skew	Negative Skew	Symmetric
Tail Direction	Right (Higher values)	Left (Lower values)	None (Balanced)
Order of Measures	Mean > Median > Mode	Mode > Median > Mean	Mean = Median = Mode
Coefficient Sign	Positive (+)	Negative (-)	Zero (0)

In a Positively Skewed distribution, the mean is greater than the median because the high-value outliers increase the sum of all values significantly without shifting the middle point (median) as much.
In a Negatively Skewed distribution, the mean is less than the median because the low-value outliers decrease the total sum, dragging the average toward the left tail.

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions