Introduction to Skewness

In Statistics, we often want to know if a set of data is spread out evenly or if it "leans" more to one side. This "lean" is called skewness. Understanding skewness helps us describe the shape of a distribution and decide which averages (like the mean or median) are the most reliable to use.

In this chapter, you will learn how to identify skewness just by looking at the data (inspection) and, for Higher tier students, how to calculate it using a specific formula.


1. What is a Symmetrical Distribution?

Before we look at skewed data, we need to know what "perfect" looks like. A symmetrical distribution is like a mirror image. If you drew a line down the middle, both sides would look the same.

In a perfectly symmetrical distribution:

\( \text{Mean} = \text{Median} = \text{Mode} \)

Example: If you measure the heights of thousands of adults, the graph usually looks like a "bell shape" where most people are in the middle, and the mean, median, and mode are all roughly the same value.


2. Skewness by Inspection (Foundation & Higher)

Identifying skewness "by inspection" means looking at the values or a diagram (like a box plot) to see which way the data is stretched. There are two main types of skewness: Positive and Negative.

Positive Skew (Right-Skewed)

In a positively skewed distribution, the data is bunched up at the low end (the left), and there is a long "tail" stretching out towards the higher values (the right).

  • The Order: \( \text{Mean} > \text{Median} > \text{Mode} \)
  • Why? The mean is "pulled" towards the high values in the long tail.
  • Box Plot Clue: The distance from the Median to the Upper Quartile (\( UQ \)) is greater than the distance from the Lower Quartile (\( LQ \)) to the Median.

Real-world example: Household income. Most people earn a similar amount, but a few billionaires create a long "tail" to the right, pulling the mean income much higher than the median.

Negative Skew (Left-Skewed)

In a negatively skewed distribution, the data is bunched up at the high end (the right), and there is a long "tail" stretching out towards the lower values (the left).

  • The Order: \( \text{Mean} < \text{Median} < \text{Mode} \)
  • Why? The mean is "pulled" down by the few very low values in the tail.
  • Box Plot Clue: The distance from the \( LQ \) to the Median is greater than the distance from the Median to the \( UQ \).

Real-world example: Scores on a very easy exam. Most students get high marks, but a few very low scores create a tail to the left.

Quick Trick to Remember: The "skew" is the direction of the tail, not the "hump." If the thin tail points to the right (positive numbers), it is positive skew. If it points to the left (negative direction), it is negative skew.


3. Using Quartiles to Identify Skew (Foundation & Higher)

When looking at a box plot or a list of quartiles, you can check for skewness by comparing the gaps between the median and the quartiles:

1. Calculate \( \text{Upper Quartile} - \text{Median} \)

2. Calculate \( \text{Median} - \text{Lower Quartile} \)

  • If \( (UQ - \text{Median}) > (\text{Median} - LQ) \), the data has a positive skew.
  • If \( (UQ - \text{Median}) < (\text{Median} - LQ) \), the data has a negative skew.
  • If the distances are roughly equal, the data is symmetrical.

4. Skewness by Calculation (Higher Tier Only)

Sometimes just looking at the data isn't enough; we need a number to tell us exactly how skewed the data is. For the Pearson Edexcel GCSE Statistics exam, you use Pearson's Coefficient of Skewness.

The formula (which is provided on the Higher tier formulae sheet) is:

\( \text{Skew} = \frac{3(\text{mean} - \text{median})}{\text{standard deviation}} \)

How to interpret the result:

  • If the answer is positive (greater than 0), the distribution has positive skew.
  • If the answer is negative (less than 0), the distribution has negative skew.
  • If the answer is 0, the distribution is symmetrical.

Example Calculation:
A dataset has a mean of \( 25 \), a median of \( 22 \), and a standard deviation of \( 5 \).
\( \text{Skew} = \frac{3(25 - 22)}{5} \)
\( \text{Skew} = \frac{3(3)}{5} = \frac{9}{5} = 1.8 \)
Since \( 1.8 \) is positive, the data has positive skew.


5. Common Mistakes to Avoid

  • Don't get "Left" and "Right" mixed up: Remember, positive skew means a tail to the right (higher values). Negative skew means a tail to the left (lower values).
  • Formula Confusion: For Higher tier students, make sure you subtract the median from the mean, not the other way around. If you swap them, your positive/negative sign will be wrong!
  • Don't ignore the Context: If an exam question asks you to "interpret" the skewness, don't just say "it's positive." Explain what that means for the data (e.g., "Most people scored low, but a few scored very high").

6. Summary Key Takeaways

For All Students:

  • Symmetrical: \( \text{Mean} = \text{Median} \). The box plot is balanced.
  • Positive Skew: \( \text{Mean} > \text{Median} \). Long tail to the right. \( UQ \) is further from the Median than the \( LQ \) is.
  • Negative Skew: \( \text{Mean} < \text{Median} \). Long tail to the left. \( LQ \) is further from the Median than the \( UQ \) is.

For Higher Tier Students:

  • Use the formula: \( \text{Skew} = \frac{3(\bar{x} - \text{median})}{\sigma} \).
  • A positive result means positive skew; a negative result means negative skew.