What is the normal distribution, and how do you use it?

Measure the mass of 1,000 apples from the same orchard. A few will be very light and a few very heavy, but most will be close to the average. If you draw a histogram of the masses and make the bars narrower and narrower, the outline turns into a smooth bell curve. That curve is the normal distribution, also called the Gaussian distribution after the mathematician Carl Friedrich Gauss.

A normal distribution is described by just two numbers:

• the mean μ, which says where the centre of the bell is;
• the standard deviation σ, which says how wide the bell is (see standard deviation).

We write X ~ N(μ, σ²). Careful: in the UK and many textbooks the second number is the variance σ², so N(60, 16) means a standard deviation of 4.

Its key properties:

• It is symmetrical about the mean, so the mean = median = mode.
• It is continuous: values can be any real number, not just whole numbers. So you find the probability of a range of values, not of one exact value, and P(X = a) = 0.
• The total area under the curve is 1, and the area between two values is the probability of landing between them.
• The curve never quite touches the x-axis, but almost all of the area is within 3σ of the mean.

The 68–95–99.7 rule (the empirical rule) gives quick estimates: about 68% of values lie within 1σ of the mean, about 95% within 2σ, and about 99.7% within 3σ.

For exact answers you standardise. The z-score says how many standard deviations a value is from the mean:

z = (x − μ) ÷ σ

The z-scores follow the standard normal distribution Z ~ N(0, 1), with mean 0 and standard deviation 1. A table (or calculator) gives Φ(z) = P(Z < z), the area to the left of z.

To work backwards (find x from a probability), look up the z that gives that area, then use x = μ + zσ.

What is the normal distribution formula?#

The curve is y = (1 ÷ (σ√(2π))) × e^(−(x − μ)² ÷ (2σ²)). At school you rarely use it directly: you standardise with z = (x − μ) ÷ σ and use a table or calculator.

How do you read the normal distribution table?#

Find the z-value to 2 decimal places: the row gives the first decimal and the column the second. The number inside is Φ(z), the probability of a value below z. For z = 1.23, go to row 1.2 and column 0.03.

How do you find normal probabilities on a calculator?#

Use the Normal CD function (normalcdf on a TI-84). Enter the lower bound, the upper bound, μ and σ (the order depends on the model). For "less than 66" use a very small lower bound such as −10⁹⁹.

How do you use the normal distribution in Excel?#

=NORM.DIST(x, mean, sd, TRUE) gives P(X < x). =NORM.S.DIST(z, TRUE) gives Φ(z), and =NORM.INV(p, mean, sd) works backwards from a probability.

Is the normal distribution continuous or discrete?#

Continuous. It models measurements such as length, mass or time, which can take any value in a range.

What is the normal distribution used for?#

To model data that cluster symmetrically around an average (heights, measurement errors, filling weights of a machine), to judge how unusual a value is, and as the basis of many tests in statistics.