Skewness and Kurtosis

 

In statistics, Skewness and Kurtosis describe the shape of a distribution curve. They are very useful in educational research, psychological testing, and classroom assessment—areas closely related to your work in education and research.

1. Skewness

Skewness measures the asymmetry of a probability distribution. It tells us if the data is shifted to one side.

Meaning

Skewness refers to the degree of asymmetry of a distribution around its mean.
If the data are not evenly distributed on both sides of the mean, the distribution is said to be skewed.

Properties of Skewness

1. Measures asymmetry of the frequency distribution.

2. In a perfectly symmetrical distribution, skewness = 0.

3. Indicates direction of deviation from normal distribution.

4. Shows the relative position of Mean, Median and Mode.

5. Helps identify extreme scores or outliers.

6. Used to check normality of data before applying parametric statistics.

Types of Skewness

1. Symmetrical Distribution

• Mean = Median = Mode

• Curve is perfectly balanced.

2. Positive Skewness (Right Skewed)

• Tail extends toward right side.

• Mean > Median > Mode.

• Many low scores and few very high scores.

3. Negative Skewness (Left Skewed)

• Tail extends toward left side.

• Mean < Median < Mode.

• Many high scores and few low scores.

• 

Formula (Conceptual)

Skewness can be calculated using Karl Pearson’s coefficient:

$$𝑆𝑘=\frac{𝑀𝑒𝑎𝑛-𝑀𝑜𝑑𝑒}{𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛}$$

or

$$𝑆𝑘=\frac{3(𝑀𝑒𝑎𝑛-𝑀𝑒𝑑𝑖𝑎𝑛)}{𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛}$$

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2. Kurtosis

Kurtosis measures the "tailedness" and peak sharpness of a distribution.

4

Meaning

Kurtosis refers to the degree of peakedness or flatness of a distribution curve compared with the normal distribution.

It shows how scores cluster around the mean.

Properties of Kurtosis

1. Indicates the shape and peak of the distribution.

2. Measures the concentration of scores around the mean.

3. Helps identify heavy tails or extreme values.

4. A normal distribution has kurtosis value ≈ 3.

5. Used in statistical analysis and test score interpretation.

Types of Kurtosis

1. Mesokurtic

• Normal distribution.

• Moderate peak.

• Kurtosis = 3.

2. Leptokurtic

• Highly peaked curve.

• Data concentrated near the mean.

• Kurtosis greater than 3.

3. Platykurtic

• Flat curve.

• Scores spread widely.

• Kurtosis less than 3.

Formula (Conceptual)

$$𝐾𝑢𝑟𝑡𝑜𝑠𝑖𝑠=\frac{𝐹𝑜𝑢𝑟𝑡ℎ𝑀𝑜𝑚𝑒𝑛𝑡}{(𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛{)}^4}$$

Uses of Skewness and Kurtosis in Teaching–Learning Process

1. Evaluation of Test Scores

Teachers can understand whether:

• Most students scored high

• Most students scored low

• Scores are normally distributed

2. Improving Question Papers

• Positive skewness → test was too difficult

• Negative skewness → test was too easy

3. Identifying Learning Differences

Helps teachers detect:

• Slow learners

• Gifted students

• Performance gaps

4. Educational Research

Used in studies related to:

• Achievement motivation

• Attitude towards ICT

• Intelligence and learning outcomes

(These types of statistical analyses are common in theses like the achievement motivation research you were working with earlier.)

5. Planning Remedial Teaching

If scores show skewness:

• Teachers can modify teaching strategies

• Provide remedial instruction

6. Validity of Statistical Tests

Researchers check skewness and kurtosis to decide whether parametric tests (t-test, ANOVA) can be applied.

Simple Summary Table

Concept	Meaning	Types
Skewness	Asymmetry of distribution	Positive, Negative, Symmetrical
Kurtosis	Peakedness of distribution	Leptokurtic, Mesokurtic, Platykurtic