The Bell Curve

The normal distribution is a continuous probability distribution that is symmetric about its mean. Its shape -- the famous bell curve -- appears in many natural phenomena: heights, test scores, measurement errors, and more.

Visual: The Normal (Bell) Curve

-3σ-2σ-1σμ+1σ+2σ+3σ

A normal distribution is fully described by two parameters: the mean (μ) which determines the centre, and the standard deviation (σ) which determines the spread.

The Empirical Rule (68-95-99.7)

For any normal distribution, the empirical rule tells us what percentage of data falls within 1, 2, and 3 standard deviations of the mean:

68%

within 1 standard deviation

μ - σ to μ + σ

95%

within 2 standard deviations

μ - 2σ to μ + 2σ

99.7%

within 3 standard deviations

μ - 3σ to μ + 3σ

Properties of the Normal Distribution

●The curve is symmetric about the mean (μ).
●The mean, median, and mode are all equal.
●The total area under the curve equals 1 (representing 100% probability).
●The curve never touches the horizontal axis (asymptotic).
●Inflection points occur at μ - σ and μ + σ.
●Exactly 50% of data lies above the mean and 50% below.

Key Vocabulary

Normal Distribution

A symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ).

Standard Deviation (σ)

A measure of spread indicating how far data values typically lie from the mean.

Empirical Rule

The rule stating that 68%, 95%, and 99.7% of data falls within 1, 2, and 3 standard deviations of the mean.

Symmetric

The left and right halves of the distribution are mirror images of each other about the mean.

Worked Examples

1

Test scores are normally distributed with μ = 70 and σ = 10. What percentage of students scored between 60 and 80?

Step 1: 60 = 70 - 10 = μ - 1σ, and 80 = 70 + 10 = μ + 1σ.

Step 2: By the empirical rule, 68% of data falls within 1 standard deviation of the mean.

Answer: Approximately 68% of students scored between 60 and 80.

2

Heights of students are normally distributed: μ = 165 cm, σ = 8 cm. What percentage are taller than 181 cm?

Step 1: 181 = 165 + 16 = 165 + 2(8) = μ + 2σ.

Step 2: By the empirical rule, 95% falls between μ - 2σ and μ + 2σ.

Step 3: So 5% falls outside this range, split equally: 2.5% above μ + 2σ.

Answer: Approximately 2.5% of students are taller than 181 cm.

3

A machine fills bottles with μ = 500 mL and σ = 5 mL. Between what values do 99.7% of the fills lie?

Step 1: 99.7% corresponds to μ ± 3σ.

Step 2: Lower bound = 500 - 3(5) = 500 - 15 = 485 mL.

Step 3: Upper bound = 500 + 3(5) = 500 + 15 = 515 mL.

Answer: 99.7% of fills lie between 485 mL and 515 mL.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

For a normal distribution, the mean, median, and mode are:

All different Mean = median, but mode is different All equal Cannot be determined

Question 2

A dataset is normally distributed with μ = 50 and σ = 6. What percentage of values fall between 38 and 62?

68% 50% 99.7% 95%

Question 3

For a normal distribution with μ = 100, σ = 15, what percentage of values are less than 100?

50% 34% 68% 95%

Question 4

Two normal distributions have the same mean but Distribution A has σ = 3 and Distribution B has σ = 8. Which is taller and narrower?

Distribution B Distribution A They look the same Cannot be determined

Question 5

Weights of apples are normally distributed with μ = 200g, σ = 25g. What percentage weigh between 175g and 225g?

50% 95% 68% 99.7%

The Normal Distribution