The Normal Distribution
Some measurements cluster in a bell-shaped pattern that a normal distribution can approximate.
Measure five hundred people and draw the histogram.
Most cluster near the middle. A few are unusually short, a few unusually tall, and the numbers thin out smoothly both ways.
Now do it with shoe sizes, or birth weights, or the errors in five hundred careful measurements.
Many such measurements have an approximately bell-shaped histogram. Others are skewed or have several peaks, so check the data before choosing a model.
That hill is the normal distribution. Once a normal model is appropriate, two numbers — the mean and standard deviation — describe that model.
Two numbers describe the whole curve
The mean μ says where the hill sits; the standard deviation σ says how wide it spreads.
Change the mean and the curve slides. Change the deviation and it gets taller and narrower, or flatter and wider.
Nothing else is needed, because the shape itself never varies.
It is a transformation again
Mean is the horizontal shift, deviation is the horizontal stretch. The same two ideas from graphing, applied to a distribution.
The 68-95-99.7 rule
About 68% of the population lies within one standard deviation of the mean, on either side.
About 95% lies within two, and about 99.7% within three. Beyond three is genuinely rare.
These three numbers answer most questions without any table or calculator.
Halve it for one side
If 95% is within two deviations, then 2.5% sits above and 2.5% below. The curve is symmetric, so split the leftover evenly.
z-scores put everything on one ruler
A z-score reports distance from the mean in standard deviations, so different measurements become comparable.
z = x - μσ. A score of z = 1.5 is one and a half deviations above average, whatever was being measured.
That lets you compare a test score with a height, which raw numbers never could.
Sign tells you which side
A negative z means below the mean. z = -2 is two deviations below, and about 2.5% of the population is further out that way.
Working backwards from a percentage
Questions often run the other way: what score puts someone in the top 16%?
Top 16% means 84% below, which is one standard deviation above the mean, since 50% + 34% = 84%.
Convert back with x = μ + zσ to get the actual score.
Draw the curve and shade
Sketching the bell and shading the region you want turns a confusing percentage into an obvious one. Almost every error here is a sketch that was never drawn.
When the bell does not apply
Not everything is normal. Incomes, city sizes and word frequencies are all skewed, with a long tail one side.
Using the empirical rule on skewed data gives confident answers that are simply wrong.
A quick histogram is the check: if it is lopsided or has two humps, the normal model does not fit.
Look before you model
The rule describes a symmetric bell. Applied to a long-tailed distribution it will badly understate how extreme the far values get.
Worked examples
Using the empirical rule
Heights are normal with mean 170 cm and deviation 8 cm. What percent are between 162 and 178 cm?
- 162 is one deviation below; 178 is one above. 170 ± 8.
- Within one deviation holds about 68%. Straight from the rule.
- So about 68%. No calculation beyond recognising the interval.
about 68%
A one-sided question
With the same heights, what percent are taller than 186 cm?
- 186 = 170 + 2(8), so z = 2. Two deviations above the mean.
- Within two deviations is 95%, leaving 5% outside. That 5% is split between the two tails.
- Half of it is above: about 2.5%. Symmetry does the splitting.
about 2.5%
Working backwards
Test scores are normal with mean 500 and deviation 100. What score is at the 84th percentile?
- 84% below means 50% + 34%, which is one deviation above. 34% is half of the middle 68%.
- So z = 1. Read off the empirical rule.
- x = 500 + 1(100) = 600. Convert the z back to a score.
600
Words to know
- Normal distribution
- The symmetric bell-shaped distribution described by its mean and standard deviation.
- Standard deviation
- The typical distance of a value from the mean — the natural ruler for spread.
- z-score
- How many standard deviations a value sits from the mean: z = x - μσ.
- Empirical rule
- About 68%, 95% and 99.7% lie within one, two and three deviations.
- Percentile
- The percentage of the population at or below a value.
Mistakes to watch for
Forgetting to halve the tail. If 95% lies within two deviations, then 5% is outside — but that is split between both ends, so only 2.5% is above. Reporting 5% doubles the answer.
Mixing up the z-score with the value. z = 2 is not a height of 2 cm; it means two deviations above the mean, which for μ = 170, σ = 8 is 186 cm. Always convert back before answering.
Applying the bell to skewed data. Incomes have a long right tail, so the empirical rule badly misjudges the top end. Check the histogram is roughly symmetric before using the model.
Dropping the sign of z. A score of z = -1.5 is below the mean, and the percentage below it is about 7%, not 93%. The sign says which side you are on.
Treating 68% as exact. It is a round figure for 68.27%, fine for estimating but not for a question demanding three decimal places — those need a table or a calculator.
It is sometimes called the Gaussian, after Gauss, who met it while studying errors in astronomical measurements.
The reason it keeps appearing has a name: the central limit theorem. Under suitable conditions, including finite variance and no dominating contribution, a standardised sum of many independent effects is approximately normal. Taking more observations does not make an arbitrary population normal.
Heights within a comparable group can be approximately normal; incomes usually are not. Income has a long tail to the right, which is why the mean income is well above the median.