Normal distribution

The normal distribution is a continuous probability distribution that is symmetrical, bell-shaped, and centred around its mean. It is one of the most important distributions in statistics because many real-world variables, such as height, weight, and test scores, tend to follow this pattern, especially when influenced by many small, random factors.

Graph of the Normal Distribution

In a normal distribution, values are most concentrated around the mean, and the probability decreases smoothly as you move further away in either direction. The distribution is fully defined by two parameters: the mean μ, which determines the centre, and the standard deviation σ, which controls the spread.

This can be represented by:

\[X\sim N\left(\mu,\sigma\right)\]

Where \(X\) is a normally distributed random variable with mean \(\mu\) and standard deviation \(\sigma\).

The peak of the curve occurs at \(x=\mu\), and the spread is determined by \(\sigma\). The most basic normal distribution, where \(\mu=0\) and \(\sigma=1\), is given by the probability density function:

\[f\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\]

Which can be plotted as:

Graph of the standard normal distribution curve, a symmetric bell-shaped curve centred at x = 0. The peak occurs at x = 0 with a maximum value near 0.4. The x-axis ranges from -3 to 3, and the y-axis shows the probability density. The curve represents the function f(x) = (1/√(2π))·e^(−x²⁄2).

This standard normal distribution can be denoted by \(Z\).  The normal distribution must satisfy several key conditions.

  • The probability can be found by finding the area under the curve.
  • The total area under the curve is always equal to 1.
  • The normal density function is bell-shaped and symmetrical, with the mean of the distribution determining the centre of the function and the standard deviation determining the width or spread of the function.

The general normal distribution then is the standard normal distribution that has been translated depending on changes to the parameters \(\mu \) and \(\sigma\). The general normal distribution is given by the function:

\[f\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]

For example, if \(\mu=1\) and \(\sigma=0.5\), the graph is translated so the centre line is now at 1 and it is the spread is more limited (compare the \(x\)-axes values with the graph above) as the data is clustered tightly around the mean.

Graph of a normal distribution curve with mean μ = 1 and standard deviation σ = 0.5. The bell-shaped curve is symmetric and peaks at x = 1 with a maximum value of approximately 0.8. The x-axis ranges from -0.5 to 2.5, and the y-axis represents probability density. The graph shows the function f(x) = (1/(σ√(2π)))·e^(−(x−μ)²⁄(2σ²)).>

If \(\mu=-1\) and \(\sigma=2\) the graph is translated so the centre line is now at -1 and the spread is wider as more of the data is further away from the mean.

Graph of a normal distribution curve with mean μ = -1 and standard deviation σ = 2. The bell-shaped curve is centred at x = -1 and is wider and lower than the standard normal curve, with a peak just above 0.2. The x-axis ranges from -7 to 5, and the y-axis represents probability density

The Standard Normal Distribution

The standard normal distribution is a powerful tool that allows us to analyse and compare data from any normal distribution using a common scale. To simplify calculations and make comparisons between different normal distributions, we often standardise the data. This process transforms any normal distribution into a standard normal distribution, which has a mean of 0 and a standard deviation of 1. Standardising allows us to apply a consistent method for analysing probabilities and interpreting values, regardless of the original scale.

When we standardise a value from a normal distribution, we convert it into a \(z\)-value, or \(z\)-score. Each \(z\)-score tells us how many standard deviations a particular data point is from the mean, and is calculated by the formula:

\[z=\frac{x-\mu}{\sigma}\]

  • A positive z-score means the value lies above the mean
  • A negative z-score means the value lies below the mean

Using z-scores, we can use standard normal values to find probabilities and percentiles without recalculating for every distribution.

Worked Example

Shane’s maths class has a \(\mu=75\) and \(\sigma=4\) for the most recent test. Shane scored 80 and wants to know how his score compared to the rest of the class.

To make this comparison, calculate the \(z\)-score for Shane's result, and state whether it is above or below the mean score for the class.

\[z=\frac{x-\mu}{\sigma}=\frac{80-75}{4}=1.25 \]

As this is a positive value, Shane’s score is above the mean. It is \(1.25\) standard deviations above the mean.

The 68-95-99.7% Rule

The 68–95–99.7% rule, sometimes called the empirical rule, is a useful guideline for understanding how data is distributed in a normal distribution. It describes how much of the data falls within one, two, and three standard deviations from the mean.

For a normally distributed random variable, approximately:

  • \(68\%\) of the values lie within one standard deviation of the mean.

Bell-shaped normal distribution curve illustrating the empirical rule. The shaded region between μ - σ and μ + σ (one standard deviation from the mean) is highlighted in green, representing 68% of the total area under the curve. The mean μ is at the centre, and the distribution is symmetric

  • \(95\%\) of the values lie within two standard deviations of the mean.

Bell-shaped normal distribution curve illustrating the empirical rule. The shaded region between μ - 2σ and μ + 2σ (two standard deviations from the mean) is highlighted in green, representing 95% of the total area under the curve. The mean μ is at the centre, and the curve is symmetric

  • \(99.7\%\) of the values lie within three standard deviations of the mean.

Bell-shaped normal distribution curve illustrating the empirical rule. The shaded region between μ - 3σ and μ + 3σ (three standard deviations from the mean) is highlighted in green, representing 99.7% of the total area under the curve. The mean μ is at the centre, and the distribution is symmetric

The 68–95–99.7% rule helps us understand how concentrated the data is around the mean in a normal distribution. Because it tells us the proportion of data within specific standard deviations, it can also be used to estimate probabilities and identify how unusual a value is. For example, if a value lies more than two standard deviations above the mean, we know it's top 2.5% of the distribution.

Normal distribution curve illustrating the empirical rule with percentage areas between standard deviations. The distribution is centred at the mean μ. Between μ ± 1σ lies 68% of the data (34% on each side), between μ ± 2σ lies 95% in total (adding 13.5% to each side), and between μ ± 3σ lies 99.7% (adding 2.35% and then 0.15% to each tail). Each segment is clearly labelled with its corresponding percentage.>

The Normal Approximation to the Binomial Distribution

The normal approximation provides a practical method for estimating binomial probabilities when direct calculation is difficult. When the number of trials, \(n\), is large and the probability of success, \(p\), is not too close to 0 or 1, the binomial distribution begins to closely resemble a normal distribution. Thus, we can use a normal approximation to the binomial distribution to simplify calculations.

When using a normal approximation to a binomial distribution, we can find the mean and standard deviation by:

\[\mu=np\] \[\sigma=\sqrt{np\left(1-p\right)}\]

For the normal approximation to the binomial distribution to be reliable, certain conditions must be met. Specifically, both the expected number of successes and failures should be sufficiently large. This is usually satisfied when \(np≥10\) and \(n(1−p)≥10\), as demonstrated in the following examples.

Three side-by-side probability mass function (PMF) plots of binomial distributions with different parameters. Each distribution is unimodal and symmetric, indicating that the probability of success p = 0.5 in all three cases. The means increase from left to right, with the left plot centred around 20, the middle around 50, and the right around 80, reflecting increasing values of n. The blue vertical lines and dots represent probabilities for each possible outcome.>