Key Statistical Concepts: Kurtosis & Hypothesis Testing

Understanding Kurtosis: Distribution Shape

Kurtosis is a statistical measure that describes the shape of a distribution’s tails compared to a normal distribution. It tells us whether the data are heavy-tailed or light-tailed.

In simple terms, kurtosis indicates the degree of peakedness and the presence of outliers in data.

Types of Kurtosis

• Mesokurtic: Normal distribution (kurtosis = 3).
• Leptokurtic: More peaked, heavy tails (kurtosis > 3).
• Platykurtic: Flatter peak, light tails (kurtosis < 3).

Key Concepts in Hypothesis Testing

1. Null Hypothesis (H₀)

It is a statistical statement that assumes no effect or no difference.

Example: “There is no difference between two groups.”

2. Alternative Hypothesis (H₁ / Hₐ)

It is the opposite of the null hypothesis. It assumes that there is an effect or difference.

Example: “There is a significant difference between two groups.”

3. Type I Error (α Error)

Rejecting the null hypothesis when it is actually true. This is often referred to as a False Positive.

4. Type II Error (β Error)

Failing to reject the null hypothesis when it is actually false. This is often referred to as a False Negative.

5. Level of Significance (α)

The probability of making a Type I error. It is the threshold set by the researcher (commonly 5% or 0.05) to decide whether to reject H₀.

Sample Size Classification and Probable Error

Sample Size Definitions

• Large Sample: If the sample size is 30 or more (n ≥ 30), it is generally considered a large sample in statistics.
• Small Sample: If the sample size is less than 30 (n < 30), it is considered a small sample.

Probable Error (P.E.)

Probable error of the mean is used to measure the reliability of the sample mean. The formula is:

$$ P.E. = 0.6745 \times \frac{\sigma}{\sqrt{n}} $$

Where:

• $\sigma$ = Standard deviation
• $n$ = Sample size

Numerical Example: Calculating Probable Error

Given Data

• Sample size ($n$) = 50
• Mean ($\bar{x}$) = 18.2 V (Note: This value is not required for P.E. calculation)
• Standard deviation ($\sigma$) = 0.2 V

Applying the Probable Error formula:

$$ P.E. = 0.6745 \times \frac{0.2}{\sqrt{50}} $$

Step 1: Calculate the Square Root of the Sample Size ($\sqrt{n}$)

$\sqrt{50} \approx 7.071$

Step 2: Divide Standard Deviation ($\sigma$) by $\sqrt{n}$

$\frac{0.2}{7.071} \approx 0.0283$

Step 3: Calculate Probable Error (Multiply by 0.6745)

$0.6745 \times 0.0283 \approx 0.0191$