Normal distribution and standard deviation - the bell curve explained step by step

The idea: most values sit close to the mean

A machine is set to fill bags with $1000$ g of flour. Weigh a thousand of its bags, and they will not all weigh exactly $1000$ g. Most will be close to it, some will be a little further off, and very few will be far off. Drawn as a histogram, the masses make a roughly symmetric hump - a bell.

The normal distribution is the mathematical model of that bell. It is set by just two numbers:

• the mean $\mu$ (the Greek letter mu) - where the middle of the bell is,
• the standard deviation $\sigma$ (sigma) - how far the values typically are from the mean. A small $\sigma$ gives a narrow, tall bell, and a large $\sigma$ a wide, low one.

The share of values in a range is the area under the curve over that range. The whole area under the curve is $1$, that is, 100%.

Why does the bell turn up so often? Roughly speaking, the sum or the mean of many independent random values is approximately normal, whatever their own distribution - that is the central limit theorem. The normal distribution is also called the Gaussian distribution, after the mathematician Carl Friedrich Gauss.

From a histogram to the bell curve

Go through the animation step by step with the buttons or the arrow keys. The bars show the masses of $1000$ bags (simulated on a computer), and the coloured curve is the normal distribution that models them.

The curve does not match every bar exactly. It is a model: it describes the pattern, not every single bag. The more bags we weigh, the closer the histogram comes to the curve - try it in the lab below.

Try it yourself: μ, σ and the area under the curve

The sliders set the machine’s mean $\mu$ and standard deviation $\sigma$, and a range from $a$ to $b$. The shaded area is the probability that a bag’s mass falls in that range. Then weigh a few hundred bags and compare the share of them in that range with the area.

What to notice:

• changing $\mu$ only moves the curve - its shape stays the same,
• a larger $\sigma$ gives a wider, lower curve, because the area under it is always $1$,
• set $a$ and $b$ one $\sigma$ either side of $\mu$ (say $\mu = 1000$, $\sigma = 4$, $a = 996$ and $b = 1004$): the area is 68.3%, whatever $\mu$ and $\sigma$ are,
• the area depends only on the z-scores of $a$ and $b$,
• the more bags you weigh, the closer the share of them from $a$ to $b$ comes to the area.

The definition

The mean and standard deviation of data. For the data $x_1, x_2, \ldots, x_n$ the mean is

The variance is the mean of the squared distances from the mean, and the standard deviation is its square root:

We square the distances so that the ones below and above the mean do not cancel out. The square root brings the result back to the units of the data, such as grams. In statistics, the standard deviation of a sample is often worked out with $n - 1$ instead of $n$ in the denominator; for large samples the difference is small.

The normal distribution. A random variable $X$ has the normal distribution with mean $\mu$ and standard deviation $\sigma$, written $N(\mu, \sigma^2)$, when its probabilities are areas under the bell curve

The curve is symmetric about $x = \mu$ and highest there. Its points of inflection, where the curve changes the way it bends, are at $\mu - \sigma$ and $\mu + \sigma$. The whole area under it is $1$. The probability that $X$ lies between $a$ and $b$ is the area under the curve from $a$ to $b$. A single value has probability $0$, so $P(a < X < b)$ and $P(a \le X \le b)$ are the same.

The 68-95-99.7 rule. For every normal distribution:

• about 68.27% of the values lie within one $\sigma$ of the mean, from $\mu - \sigma$ to $\mu + \sigma$,
• about 95.45% lie within $2\sigma$,
• about 99.73% lie within $3\sigma$.

Standardising. The z-score $z = \frac{x - \mu}{\sigma}$ says how many standard deviations $x$ is from the mean. If $X$ has the distribution $N(\mu, \sigma^2)$, then $Z = \frac{X - \mu}{\sigma}$ has the standard normal distribution $N(0, 1)$. Its areas $\Phi(z) = P(Z < z)$ are in tables and on calculators:

Many tables list only $z \ge 0$. For a negative $z$, use the symmetry of the curve:

For a range from $a$ to $b$, with $z_a$ and $z_b$ the z-scores of its ends:

Example 1: the standard deviation of five numbers

Five students timed how long a puzzle took them, in minutes: $4$, $6$, $7$, $8$ and $10$. Find the mean and the standard deviation of these times.

Example 2: the share of light bags

The mass of a bag of flour follows a normal distribution with mean $\mu = 1000$ g and standard deviation $\sigma = 5$ g. What share of the bags weigh less than $992$ g?

Your turn

The heights of young tomato plants in a greenhouse follow a normal distribution with mean $60$ cm and standard deviation $12$ cm. What share of the plants are between $54$ cm and $78$ cm tall?

Common mistakes

• Looking up $x$ instead of $z$. The table is for the standard normal distribution, so standardise first. For $992$ g with $\mu = 1000$ g and $\sigma = 5$ g, work with $z = -1.6$, not with $992$.
• Forgetting the “1 −”. $\Phi(z)$ is the area to the left of $z$. For “more than” you need the area to the right: $P(Z > z) = 1 - \Phi(z)$. For example, $P(Z > 1) = 1 - 0.8413 = 0.1587$.
• Mixing up $\sigma$ and $\sigma^2$. Books write the normal distribution either as $N(\mu, \sigma^2)$ or as $N(\mu, \sigma)$. In $N(10, 16)$, written the first way, the standard deviation is $\sqrt{16} = 4$, not $16$. Check which way your book writes it.
• Using 68-95-99.7 for any data. The rule holds for normal distributions. For other shapes the shares are different: in a uniform distribution, where every value in an interval is equally likely, only about 58% of the values lie within one $\sigma$ of the mean.
• “A larger σ makes the curve taller.” It is the other way round. The area under the curve is always $1$, so a wider curve must be lower: the height of its peak is $\frac{1}{\sigma\sqrt{2\pi}}$.

In exams and at university

In England, the normal distribution is part of A-level Mathematics, though not of the shorter AS course. Students use it as a model and find probabilities from it, and they link it to histograms, the mean, the standard deviation and the points of inflection. They must be able to get probabilities from statistical distributions such as this one on a calculator. A-level writes the distribution as $N(\mu, \sigma^2)$, with the variance as the second number. Students also test hypotheses about the mean of a normal distribution. The standard deviation itself comes earlier, already in the AS content: students calculate and interpret it, including from summary statistics.

In the United States, the normal distribution is a topic of AP Statistics, a course equivalent to a one-semester introductory college course in statistics.

At university, the normal distribution is a standard topic of a first course in statistics. In the open textbook OpenIntro Statistics it comes after the chapters on summarising data and on probability. The next chapter, on the foundations of inference, builds on it and introduces the central limit theorem.

Check yourself

Flashcards for review

Flip a card and say whether you knew the answer. The ones you did not know come back at the end of the deck. Why it pays to come back to them in a few days is explained in the guide Spaced repetition.