The Normal Distribution & Visualizer

A probability distribution is a combination of a variable and its corresponding probabilities of each value. The Normal Distribution, known as the bell curve, is one of the most significant statistical distributions in use today. Its typical nature is explained in the following diagram:

Most human characteristics, such as height, weight, waist-size, blood pressure, etc., follow a normal distribution. The mean of the distribution is located at the center of the curve, while the standard deviation determines the spread of the data. The area under the curve represents the total probability, which is equal to 1 that makes it a probability distribution.
The normal distribution is widely used in various fields, including statistics, finance, and social sciences, to model real-world phenomena and make predictions based on data.

The smaller the standard deviation, the narrower and taller the bell curve will be, indicating that the data points are closely clustered around the mean. On the other hand, a larger standard deviation results in a wider and flatter curve, suggesting that the data points are more spread out from the mean. The normal distribution is also characterized by its symmetry around the mean, meaning that the left and right sides of the curve are mirror images of each other. This property allows for various statistical analyses and hypothesis testing to be conducted using the normal distribution as a reference point.
In addition, mode and median are also located at the center of the curve, which means that in a normal distribution, the mean, median, and mode are all equal. This is a unique property of the normal distribution and contributes to its widespread use in statistical analysis.

Properties of the Normal Distribution

• 💎 Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the curve.
• 💎 Symmetry: The normal distribution is symmetric around the mean, meaning that the left and right sides of the curve are mirror images of each other.
• 💎 68-95-99.7 Rule: Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

With the following simulation, you can visualize the normal distribution: choose your value of standard deviation and mean with the sliders below and adjust the size of the variable to see the area under the curve.

 

Worked Examples

E.g.1

The diameters of a nail produced by a particular machine, Xmm, are modelled as X~N(6, 0.5). Find: a) P(X > 6)
b) P(5.5 < X <6.5)
c) P(X < 5.5)
d) P(X > 7)
e) P(5 < X < 7)

Open Simulation

Answers

1. a) P(X > 6) = 0.5
2. b) P(5.5 < X < 6.5) = 0.6827
3. c) P(X < 5.5) = 0.1587
4. d) P(X > 7) = 0.0228
5. e) P(5 < X < 7) = 0.9545

E.g.2

The arm spans of a group of teenagers in a gymnastic club, Xcm, are modelled as X ~ N(120, 9). a State the proportion of students who have an arm span between 117 cm and 123 cm. b State the proportion of students who have an arm span between 114 cm and 126 cm.

Answers

1. a) P(117 < X < 123) = 0.68 (approx — using the 68% rule, ±1σ)
2. b) P(114 < X < 126) = 0.95 (approx — using the 95% rule, ±2σ)

E.g.3

The lengths of a group of cats, X cm, are modelled as X ~ N(30, σ²). If 68% of the cats have a length between 23 cm and 37 cm, find the variance, σ2.

Answer

Using the 68% rule (≈ mean ± 1σ): the interval 23 to 37 is 30 ± σ, so σ = 7. Therefore the variance is σ² = 49.

E.g.4

The weights of a group of dogs, X kg, are modelled as X ~ N(20, σ²). If 95% of the dogs have a weight between 10 kg and 30 kg, find the variance, σ2.

Answer

Using the 95% rule (≈ mean ±2σ): the interval 10 to 30 is 20 ± 2σ, so σ = 5. Therefore the variance is σ² = 25.

E.g.5

The weights of a group of squirrels , Q grams, are modelled as Q ~ N(μ, 5²). If 97.5% of the squirrels weigh less than 120 grams, find the mean.

Answer

Using the 97.5% rule (≈ mean + 2σ): the interval is Q < 120, so 120 = μ + 2(5) → μ = 110 grams. Due to the symmetry of the bell curve. the area of the two extremes are 2.5% each, that leaves the area of the central section at 95%.

E.g.6

The masses of a herd of cows, M kg, on a farm are modelled as M ~ N(μ, σ²). If 84% of the cows weigh more than 180 kg and 97.5% of the cows weigh more than 160 kg, find µ and σ.

Answer

Using the 68% rule: about 16% lies below μ − σ. Here 84% weigh more than 180 kg, so 16% weigh ≤ 180 kg → 180 ≈ μ − σ.

Using the 95% rule: about 2.5% lies below μ − 2σ. Here 97.5% weigh more than 160 kg, so 2.5% weigh ≤ 160 kg → 160 ≈ μ − 2σ.

Solve the two equations: μ − σ = 180 and μ − 2σ = 160. Subtracting gives σ = 20 kg, then μ = 200 kg standard deviation = 20 kg.