standard deviation

The normal curve (bell curve) is one of the most important shapes in statistics. It appears when many small, independent factors combine: height, test scores, measurement errors.

For a simple, intuitive presentation go to Part 2

Properties of the Normal Curve

1. Symmetrical around the mean
2. Unimodal (one peak)
3. Mean = Median = Mode
4. The total area under the curve = 1 (or 100%)

Formula for the Normal Distribution

Unless you are in Mathematical Statistics, you will never be asked to reproduce it, or otherwise work with it.

Symbolic formula:
$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$

Formula in words:
$$\text{Probability density} = \frac{1}{\text{standard deviation} \times \sqrt{2\pi}} \times e^{-\frac{(\text{score} - \text{mean})^2}{2 \times (\text{standard deviation})^2}}$$

Where:

• $$\mu$$ = mean
• $$\sigma$$ = standard deviation
• $$x$$ = a value on the curve

Standardization (z-scores)

Symbolic formula:
$$z = \frac{x - \mu}{\sigma}$$

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

A z-score tells us how many standard deviations a score is above or below the mean.

Key Percentages

Under the normal curve:

• About 68% of scores are within 1 standard deviation of the mean
• About 95% are within 2 standard deviations
• About 99.7% are within 3 standard deviations

This is called the 68–95–99.7 rule.

Example

Suppose test scores are normally distributed with

• $$\mu = 100$$
• $$\sigma = 15$$

What is the z-score for a student who scored 115?

$$z = \frac{115 - 100}{15} = \frac{15}{15} = 1$$

This means the student is 1 standard deviation above the mean.

Visuals

Figure 4.1 — The Normal Curve. A bell-shaped curve centered at the mean (μ).

Figure 4.2 — The 68–95–99.7 Rule. A normal curve with shaded regions for ±1σ, ±2σ, ±3σ.

Figure 4.3 — z-Score Example. Normal curve with shaded area to the left of z = 1.0, labeled 0.8413.

Why This Matters

The normal curve is the foundation of inferential statistics.

• It allows us to compute probabilities.
• It underlies the t-test, ANOVA, and confidence intervals.
• By using z-scores, we can compare scores across different tests and distributions.

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After finding the mean, the next question is: How much do the scores vary around that mean?
Variation tells us whether data are tightly clustered or widely spread. Two common measures are the variance and the standard deviation.

Variance and standard deviation - formal level

Variance and standard deviation - intuitive level

Variance

Variance is the average squared distance of each score 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

Example: Data: 6, 8, 10

• Mean = 8
• Deviations: (6–8) = –2, (8–8) = 0, (10–8) = 2
• Squared deviations: 4, 0, 4
• Sum = 8

Variance = $$\tfrac{8}{3-1} = 4$$

Standard Deviation

The standard deviation is the square root of the variance.

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 continued:
Variance = 4 → Standard deviation = $$\sqrt{4} = 2$$

So, on average, scores are about 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 3.1 — Variability Around the Mean. A dot plot of scores with the mean marked, vertical lines showing deviations, and shaded boxes for squared deviations.

Why This Matters

Two sets of data can have the same mean but very different spreads.
Variance and standard deviation give us the language to describe that spread.
They are the foundation for most inferential tests in statistics.

Practice self-test quiz

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