The normal distribution
Lengths of needles, heights of people, errors in a careful measurement all pile up in the same symmetric hump. Is there one curve for all of them, and what does it take to pin it down?
Where you are 8 in Distributions
Why this exists
Measure the length of a few thousand needles from one pine tree and draw a histogram. Most needles are close to some typical length, fewer are a little longer or shorter, and very few are far off in either direction. The histogram is a single hump, roughly symmetric, falling away smoothly on both sides. The same hump turns up for the heights of adults of one sex, the masses of coins fresh from a mint, and the errors when one quantity is measured over and over with care.
A density curve can model any of these, but there are endless curves to choose from. What makes this shape so useful is that one family of curves fits all of these humps, and a curve in that family is fixed completely by just two numbers: where its center is and how spread out it is. Those are the mean and the Standard deviation. Once you know them, you know the whole curve, and with it every proportion the model gives. That family is the normal distribution, and it is the model most of statistics is built on.
Definition
A normal distribution is a model for a numerical variable whose density curve is a symmetric, single-peaked bell. Each normal curve is fixed by two numbers:
- its mean \mu, the center, where the peak stands, and
- its standard deviation \sigma, a positive number giving how far the values typically spread from the mean.
The normal distribution with mean \mu and standard deviation \sigma is written N(\mu, \sigma). Because these two numbers settle everything about the curve, they are called its parameters.
For reference, the curve is the graph of
where e \approx 2.718 is the base of the The number e and the natural exponential function. Nothing in this lesson needs you to work with this formula directly; it is here so that the facts below have something exact behind them.
A curve that bends downward, like the top of a hill, is concave down; one that bends upward, like the bottom of a bowl, is concave up. A point where a curve changes from one to the other is an inflection point.
Intuition
One hump, one center. The bell is highest at the mean and falls away the same way on both sides, so the mean is also the median (half the area on each side) and the mode (the highest point). On a skewed curve these three come apart; on a normal curve they are one point.
Where to see the standard deviation. Near the peak the bell is shaped like the top of a hill, bending downward. Far out it is like the foot of a hill, bending upward as it flattens toward the axis. The change from one to the other happens at exactly one standard deviation either side of the mean. So the standard deviation is not only a number from a formula: you can see it on the graph, as the distance from the peak’s center line to either inflection point.
The figure shows the model N(42, 3) for the lengths of the pine needles, in millimeters. Its peak is above 42 and its inflection points are above 39 and 45. By 33 and 51, three standard deviations out, the curve is so close to the axis that it can hardly be told apart from it, which is why a normal curve is usually drawn from \mu - 3\sigma to \mu + 3\sigma.
What each parameter does. Change \mu and the whole curve slides left or right, keeping its shape exactly. Change \sigma and the curve stretches or squeezes sideways about its center. A density curve must keep total area 1, so a curve stretched to twice the width must come down to half the height. A small \sigma means a tall, narrow bell, values packed close to the mean; a large \sigma means a low, wide one.
The tails never quite stop. The curve gets closer and closer to the axis but never touches it, so in the model every value, however far from the mean, has some area near it. The area that far out is tiny, which is why the model can still describe real measurements that do have limits.
Results
Symmetry. A normal curve is symmetric about the vertical line x = \mu. Its mean, median and mode are all \mu, and the share of the area on either side of \mu is \displaystyle \frac{1}{2}. For any distance d, the share above \mu + d equals the share below \mu - d.
The peak. The highest point is above x = \mu, at height
So doubling \sigma halves the height of the peak.
Inflection points. The curve is concave down between \mu - \sigma and \mu + \sigma and concave up outside them. Its two inflection points are above x = \mu - \sigma and x = \mu + \sigma.
Two parameters, one shape. Every normal curve is a copy of every other, slid sideways and stretched. Changing \mu slides the curve; changing \sigma stretches it by the same factor sideways and squashes it by that factor upwards.
Changing units. If a variable has distribution N(\mu, \sigma) and every value is changed to a + bx with b \neq 0, as in a change of units, the new values have distribution
Adding a constant moves the mean and leaves the spread alone; multiplying by b multiplies both the mean and the spread, the same rules as for any data set (shifting and rescaling a data set). What is special here is that the result is still normal.
Why it is true
Symmetry. The formula depends on x only through (x - \mu)^2. A value a distance d above the mean and a value d below give the same square, d^2, so the curve has the same height at \mu + d and \mu - d. A curve with mirror symmetry has its median and mean on the mirror line, as Density curves: area, mean and median showed, so both are \mu.
The peak. The power of e is \displaystyle -\frac{(x - \mu)^2}{2\sigma^2}. A square is never negative, so this power is 0 at x = \mu and negative everywhere else. Since e^0 = 1 and e^t < 1 for negative t, the height is greatest at x = \mu, where it is \displaystyle \frac{1}{\sigma\sqrt{2\pi}}. The further x is from \mu, the more negative the power and the lower the curve; but a power of e is never 0, so the curve never reaches the axis.
One shape, slid and stretched. Write \displaystyle z = \frac{x - \mu}{\sigma}, the number of standard deviations x is from the mean. Then
The second factor is the same curve for every \mu and \sigma: the standard normal curve, as a function of z. Going from z to x = \mu + \sigma z slides it to \mu and stretches it sideways by \sigma, which multiplies every area by \sigma; the factor \displaystyle \frac{1}{\sigma} in front squashes it upwards by the same amount and brings the area back to 1. That is why a wider normal curve is a lower one.
Inflection points. Bending is about how the slope changes, which calculus measures with the second derivative (Concavity and inflection points). For g(z) = e^{-z^2/2} the second derivative is
The power of e is always positive, so the sign is the sign of z^2 - 1: negative for -1 < z < 1 (concave down) and positive outside (concave up). The change happens at z = \pm 1. Sliding and stretching carry z = \pm 1 to x = \mu \pm \sigma.
Area 1 and the meaning of \sigma. That the area under the curve is exactly 1, and that the mean and standard deviation of the model really are \mu and \sigma, are facts from calculus; the \displaystyle \sqrt{2\pi} in the formula is the number that makes the area come out to 1.
Changing units. The new values are a + b(\mu + \sigma z) = (a + b\mu) + (b\sigma)z, a slide and a stretch of the same bell again, so the result is normal. Its center is a + b\mu, and its spread is |b|\sigma because a standard deviation is a distance and cannot be negative; a negative b also flips the curve over, which a symmetric curve does not notice.
Worked examples
Sketching a normal curve
The lengths of the needles on one pine tree are modeled by N(42, 3), in millimeters. Describe the curve: where its peak is, where its inflection points are, over what range it is worth drawing, how tall it is, and what share of needles the model says are shorter than 42 mm.
SolutionThe peak is above the mean, 42 mm, because a normal curve is highest at \mu.
The inflection points are one standard deviation either side: 42 - 3 = 39 mm and 42 + 3 = 45 mm. Between them the curve bends downward; outside them it bends upward and flattens out.
Three standard deviations either side runs from 42 - 9 = 33 to 42 + 9 = 51 mm. Beyond these the curve is almost on the axis, so the sketch runs from 33 to 51.
The peak height is
That is a density, a share per millimeter, not a share.
By symmetry, half the area lies on each side of the mean, so the model says \displaystyle \frac{1}{2} of the needles are shorter than 42 mm. This is the curve in the first figure.
Reading the parameters off a graph
The time a bag of microwave popcorn takes to finish popping is modeled by a normal curve. Its peak is above 160 seconds, and it changes from bending down to bending up above 152 seconds and above 168 seconds. Find \mu and \sigma, and write the model.
SolutionThe peak of a normal curve stands above its mean, so \mu = 160 seconds.
The inflection points sit one standard deviation from the mean. The distance from 160 to 168 is 8, and from 152 to 160 is also 8, as it must be for a symmetric curve. So \sigma = 8 seconds, and the model is N(160, 8).
As a check, the mean is also the midpoint of the two inflection points: \displaystyle \frac{152 + 168}{2} = 160, and their distance apart, 168 - 152 = 16, is 2\sigma.
Comparing three curves
Three variables are modeled by A = N(50, 5), B = N(50, 10) and C = N(65, 5). Which curves have the same center, which have the same shape, and which is tallest? Give each curve’s peak height and inflection points.
SolutionA and B have the same mean, 50, so their peaks stand over the same point. A and C have the same standard deviation, 5, so they are the same shape: C is A slid 15 to the right.
The peak height is \displaystyle \frac{1}{\sigma\sqrt{2\pi}}, which depends only on \sigma:
A and C are equally tall, and B is half their height, because its standard deviation is twice theirs: twice as wide, so half as tall, keeping area 1.
The inflection points are at \mu \pm \sigma: 45 and 55 for A, 40 and 60 for B, and 60 and 70 for C. The second figure shows all three.
From trip length to fare
The lengths of trips taken with one taxi company are modeled by N(4, 1.5) in miles. A trip of m miles costs 3 + 2m dollars. What is the model for the fares? Where are its inflection points, and what happens to the peak height?
SolutionChanging every value to 3 + 2m gives a normal distribution with mean 3 + 2 \times 4 = 11 and standard deviation 2 \times 1.5 = 3, so the model is N(11, 3) in dollars. Adding 3 moves the center and has no effect on the spread; multiplying by 2 stretches both.
The inflection points are at 11 - 3 = 8 and 11 + 3 = 14. These are exactly the fares for the trips at the inflection points of the length curve, 2.5 and 5.5 miles: 3 + 2 \times 2.5 = 8 and 3 + 2 \times 5.5 = 14. Rescaling does not move the curve’s features, only the numbers written under them.
The peak height falls from
A dollar of fare covers only half a mile of trip, so the share of trips per dollar is half the share per mile. The shape on the page is the same; only the scale has changed.
Using symmetry
For the pine needles, N(42, 3), the model says a share of about 0.091 of needles are longer than 46 mm. What share are shorter than 38 mm? What share are between 38 and 46 mm? What share are between 42 and 46 mm?
Solution46 is 4 above the mean and 38 is 4 below it. A normal curve is symmetric about its mean, so the tail below 38 is the mirror image of the tail above 46 and has the same area: about 0.091.
The whole area is 1, so the share between 38 and 46 is what is left after both tails:
Half the area lies above 42, and 0.091 of it lies above 46, so the share between 42 and 46 is about 0.5 - 0.091 = 0.409.
Finding a share like 0.091 in the first place, from \mu and \sigma alone, is the work of normal calculations. Symmetry and total area 1 turn one such share into several.
When a normal model fits and when it does not
Decide whether a normal model is reasonable for each variable.
(a) Amounts spent by customers at a café, which have mean \$6 and standard deviation \$5. (b) Times taken by the workers in a large city to travel to work, which have mean 27 minutes and median 24 minutes. (c) Diameters of ball bearings made by one machine, which have mean 8.00 mm and standard deviation 0.02 mm, with a histogram that is a single symmetric hump.
Solution(a) No. An amount spent cannot be negative, yet 0 is only \displaystyle \frac{6}{5} = 1.2 standard deviations below the mean. A normal curve with these parameters puts a sizeable piece of its area below 0, about 0.115, so the model would say roughly one customer in nine spends less than nothing. Data that pile up against a hard floor like this, with a standard deviation nearly as large as the mean, are usually skewed to the right: many small bills and a long tail of large ones.
(b) Doubtful. On a normal curve the mean and median are equal. Here the mean is 3 minutes above the median, a sign of a long right tail (a few very long journeys pull the mean up, as in Mean, median and skew: choosing the center). A normal model would place the middle of the times in the wrong place.
(c) Reasonable. The histogram has the single symmetric hump a normal curve has. The diameter cannot be negative, but 0 is \displaystyle \frac{8.00}{0.02} = 400 standard deviations below the mean, where the curve’s area is far too small to matter.
A normal model is a choice to be checked against the data, never assumed; checking it carefully, with a plot made for the purpose, is Checking whether data look normal.
Boundaries
A model, not a fact about the data. Real measurements have limits (no needle is longer than the tree) and come in recorded steps, while the normal curve runs on for ever in both directions and is smooth. No real variable is exactly normal. The question is always whether the model is close enough for the purpose, and that has to be checked, not assumed.
Symmetric, single-peaked shapes only. A clearly skewed distribution, one with two peaks (the heights of a mixed group of children and adults, say), or one piled against a floor or a ceiling is not normal, and no choice of \mu and \sigma will make the curve fit it. Not every symmetric shape is normal either: a flat or triangular shape is symmetric but has the wrong form.
\sigma must be positive. A standard deviation of 0 would squeeze every value onto the mean, and the curve would be infinitely tall and infinitely thin, which is not a density curve.
The height of the curve is not a share. As with any density curve, only area gives a share. A normal curve with a small enough \sigma has a peak taller than 1.
Notation varies. This lesson writes N(\mu, \sigma) with the standard deviation second. Many books and calculators write N(\mu, \sigma^2), with the variance second instead, so N(42, 9) there means the same curve as N(42, 3) here. Read which one a source uses before taking the second number.
Where this goes next
Every normal curve is the standard one slid and stretched, so measuring a value in standard deviations from the mean, its z-score, puts all normal curves on one scale. The same lesson gives the empirical rule, the rough shares of a normal curve’s area within one, two and three standard deviations of the mean. The standard normal distribution, N(0, 1), is the single curve they all become, and normal calculations read areas and percentiles from it, turning “about 0.091 of needles are longer than 46 mm” from a given into something you can find. Checking normality with a plot decides whether the model is reasonable in the first place.
Beyond that, the normal curve is the shape of a normal random variable, and the The central limit theorem explains why it appears so often: an average of many independent small effects comes out close to normal whatever the effects look like. That is what makes it the backbone of Sampling distributions and of inference.
Exercises
Reading the parameters
- A normal curve has its peak above 120 and inflection points above 112 and 128. Find \mu and \sigma.
- A variable is modeled by N(75, 6). Where are the inflection points of its curve, and what is its median?
- A normal curve has inflection points above 2.4 and 3.0. Find its mean and standard deviation.
- A normal curve is centered at 18 and has one inflection point above 22.5. Where is the other, and what is \sigma?
- Over what range of values is it sensible to draw the curve of N(250, 20), running three standard deviations either side of the mean?
Height and width
- Find the peak heights of N(10, 2) and N(30, 4), to three significant figures. How do they compare, and why?
- A normal curve has a peak height of 0.04. Find its standard deviation, to three significant figures.
- Three curves are P = N(40, 5), Q = N(40, 2) and R = N(55, 5). Which is tallest? Which two have the same shape? Which two share a center line?
- The curve N(30, 2) is replaced by N(30, 6). Find the peak height of each, to three significant figures, and the inflection points of each.
Symmetry
- A variable is modeled by N(60, 4), and the model gives a share of about 0.067 above 66. What share lies below 54, and what share between 54 and 66?
- For the same model, N(60, 4), the share below 57 is about 0.227. What share lies above 63? What share lies between 60 and 63?
- A normal model gives the same share below 30 as above 50. What is its mean?
Changing units
- Night-time temperatures are modeled by N(15, 2.5) in degrees Celsius. Using 1.8c + 32, give the model in degrees Fahrenheit.
- Times to finish a puzzle are modeled by N(90, 12) in seconds. Give the model in minutes.
- The masses of parcels are modeled by N(2.4, 0.3) in kilograms. Each is packed in a box weighing 0.5 kg. Give the model for the packed masses.
Is a normal model reasonable?
- House prices in a city have mean \$310{,}000 and median \$250{,}000. Is a normal model reasonable? Explain.
- Hours of overtime worked in a week by the staff of a firm have mean 4 and standard deviation 5, and no one works negative overtime. A normal model would give a share of about 0.21 below 0. Is a normal model reasonable?
- The masses of bags of flour from one machine have mean 1000 g and standard deviation 5 g, and a histogram with one symmetric hump. Masses cannot be negative. Does that rule out a normal model?
- A group is made up of six-year-olds and their parents. Would a normal model suit the heights of the whole group? Explain.
Explaining
- Explain why the mean, median and mode of a normal distribution are all the same number.
- Explain why two normal curves with different standard deviations cannot have peaks of the same height.
- The pine needle lengths were modeled by N(42, 3). Explain why no normal model can be exactly right for lengths of needles, and why the model can still be useful.
Answers
- \mu = 120, \sigma = 8. The peak is above the mean, and each inflection point is 128 - 120 = 8 from it.
- Above 69 and 81, one standard deviation either side of 75. The median is 75, the same as the mean, by symmetry.
- \mu = 2.7, \sigma = 0.3. The mean is the midpoint \displaystyle \frac{2.4 + 3.0}{2} = 2.7, and each point is 0.3 from it.
- Above 13.5, and \sigma = 4.5. The inflection point is 22.5 - 18 = 4.5 above the mean, so the other is 18 - 4.5 = 13.5.
- From 190 to 310: 250 - 3 \times 20 = 190 and 250 + 3 \times 20 = 310.
- About 0.199 and 0.0997. \displaystyle \frac{1}{2\sqrt{2\pi}} \approx 0.199 and \displaystyle \frac{1}{4\sqrt{2\pi}} \approx 0.0997. The first is twice the second: its standard deviation is half as big, so the curve is half as wide and must be twice as tall to keep area 1. The means play no part.
- \sigma \approx 9.97. From \displaystyle \frac{1}{\sigma\sqrt{2\pi}} = 0.04, \displaystyle \sigma = \frac{1}{0.04\sqrt{2\pi}} = \frac{25}{\sqrt{2\pi}}.
- Q is tallest, having the smallest \sigma. P and R have the same shape (both \sigma = 5); R is P slid 15 to the right. P and Q share the center line x = 40.
- N(30, 2): peak about 0.199, inflection points above 28 and 32. N(30, 6): peak \displaystyle \frac{1}{6\sqrt{2\pi}} \approx 0.0665, a third as tall, with inflection points above 24 and 36, three times as far out.
- About 0.067 below 54, by symmetry, since 54 and 66 are each 6 from the mean. Between them: 1 - 0.067 - 0.067 = 0.866.
- About 0.227 above 63, the mirror image of the tail below 57. Between 60 and 63: 0.5 - 0.227 = 0.273, since half the area is above the mean.
- \mu = 40. Equal tails on the two sides means the two values are mirror images in the mean, so the mean is their midpoint, \displaystyle \frac{30 + 50}{2} = 40.
- N(59, 4.5): mean 1.8 \times 15 + 32 = 59 and standard deviation 1.8 \times 2.5 = 4.5, in degrees Fahrenheit.
- N(1.5, 0.2): dividing by 60 gives mean \displaystyle \frac{90}{60} = 1.5 and standard deviation \displaystyle \frac{12}{60} = 0.2 minutes.
- N(2.9, 0.3): adding 0.5 moves the mean to 2.9 and leaves the standard deviation at 0.3.
- No. A normal distribution has mean equal to median; here the mean is far above the median, a sign of a long right tail of very expensive houses. The prices are skewed right.
- No. A model that puts about a fifth of its area on impossible values is a poor fit; with a floor at 0 and \sigma larger than \mu, the hours are almost certainly skewed right, with many staff at or near 0.
- No. 0 is \displaystyle \frac{1000}{5} = 200 standard deviations below the mean, and the area of a normal curve that far out is too small to matter. With a single symmetric hump, a normal model is reasonable.
- No. The heights form two clusters, one for the children and one for the adults, so the distribution has two peaks. A normal curve has only one.
- The curve is symmetric about x = \mu, so half the area lies on each side of it, making \mu the median, and the balance point is on the mirror line, making \mu the mean. The highest point of the curve is also above x = \mu, so \mu is the mode.
- The peak height is \displaystyle \frac{1}{\sigma\sqrt{2\pi}}, which depends only on \sigma and gets smaller as \sigma gets larger. Different values of \sigma give different heights. In picture form: the wider curve must be lower to keep its area equal to 1.
- The normal curve has area, however small, over every number, including negative lengths and lengths longer than any pine needle, and it treats length as perfectly smooth, while real measurements are rounded. So no normal curve matches the needles exactly. The model is still useful because the area it puts on impossible lengths is tiny (0 is 14 standard deviations below 42), and for the values that do occur it can describe the shares closely, which is what it is used for.