The normal distribution is not just a shape — it is a powerful tool.
It allows us to describe data, calculate probabilities, and make decisions about means and differences.

Here are four major uses of the normal curve.

1. Describing Data

The normal curve summarizes how scores are distributed.

• Mean = center
• Standard deviation = spread

It provides a reference point: where most scores fall, and where extremes occur.

Figure L4.1 — Normal Curve with mean and ±1σ, ±2σ, ±3σ marked.

2. Probability of a Score

We can use the normal curve to calculate the probability of observing a score above or below a certain value.

Formula for standardization:
$$z = \frac{x - \mu}{\sigma}$$

Formula in words:
$$z = \frac{\text{score} - \text{mean}}{\text{standard deviation}}$$

The z-score tells us how many standard deviations a score is from the mean.
With the z-table, we can find the probability of that score.

Figure L4.2 — Normal curve with shaded area above z = 1.5.

3. Reliability of a Mean (SEM)

If we take many samples, the means vary. The Standard Error of the Mean (SEM) tells us how much.

Formula:
$$\mathrm{SEM} = \frac{s}{\sqrt{n}}$$

Formula in words:
$$\text{SEM} = \frac{\text{standard deviation}}{\sqrt{\text{number of scores}}}$$

Smaller SEM means the sample mean is a more reliable estimate of the population mean.

Figure L4.3 — Distribution of sample means, narrower than distribution of raw scores.

4. Reliability of a Difference

The normal distribution also underlies hypothesis testing — such as the t-test.
It allows us to compare two means and decide whether their difference is larger than expected by chance.

Figure L4.4 — Two overlapping normal curves with different means.

Why This Matters

The normal distribution is the foundation for:

• Calculating probabilities
• Estimating reliability of means
• Testing hypotheses about differences

Understanding these uses prepares us for the transition from descriptive to inferential statistics.

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.

The mean tells us the “typical” score. But how tightly do scores cluster around the mean? Do they spread widely, or are they close together?

To answer, we measure variability. Two key measures are the variance and the standard deviation.

Variance

Variance is the average squared distance of scores from the mean.

Symbolic formula:
$$s^2 = \frac{\sum (X - \bar{X})^2}{n - 1}$$

Formula in words:
$$\text{Variance} = \frac{\text{sum of squared deviations from the mean}}{\text{number of scores} - 1}$$

Where:

• $$s^2$$ = variance
• $$X$$ = each score
• $$\bar{X}$$ = mean
• $$n$$ = number of scores

Standard Deviation

The standard deviation is the square root of the variance. It puts variability back into the same units as the data.

Symbolic formula:
$$s = \sqrt{\frac{\sum (X - \bar{X})^2}{n - 1}}$$

Formula in words:
$$\text{Standard deviation} = \sqrt{\frac{\text{sum of squared deviations from the mean}}{\text{number of scores} - 1}}$$

Example

Data: 6, 8, 10

• Mean = 8
• Deviations: –2, 0, 2
• Squared deviations: 4, 0, 4
• Sum = 8

Variance:
$$s^2 = \frac{8}{3-1} = 4$$

Standard deviation:
$$s = \sqrt{4} = 2$$

So, on average, scores are 2 units away from the mean.

Definition

• Variance: average squared distance from the mean.
• Standard Deviation: square root of variance; typical distance from the mean.

Visuals

Figure L3.1 — Variability Around the Mean. Dot plot of scores with the mean marked, vertical lines for deviations, and shaded boxes for squared deviations.

Why This Matters

Two sets of scores can have the same mean but very different spreads.
Variance and standard deviation give us the language to describe spread, and they are the building blocks for t-tests, ANOVA, and all inferential statistics.

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.