Coaching Centre
Normally distributed data
What normally distributed data looks like, the properties of the normal curve, real examples, and how to recognise data that is not normal.
Most Year 12 boys are close to the average height. A few are quite a bit taller or shorter, and very few are extremely tall or short. When lots of small, random effects add together (genes, diet, sleep), you get this pattern: most values bunch in the middle and they thin out evenly on each side.
The bell curve
When a histogram of a large dataset has a symmetric, bell-shaped outline, we say the data is normally distributed. The smooth curve drawn over it is the normal curve.
| Property | What it means |
|---|---|
| Bell-shaped | One peak in the middle, tails on both sides |
| Symmetric about the mean | The left half is a mirror image of the right half |
| Mean $=$ median $=$ mode | All three sit at the centre, under the peak |
| Most data near the centre | Values far from the mean are rare |
| Fixed by two numbers | The mean sets the centre, the standard deviation sets the spread |
Normal or not?
Many measurements are close to normal: heights, the mass of packaged products, the volume a machine pours into bottles, reaction times in a large group, and marks on a well-designed exam. Other data is clearly not:
- Positively skewed (long tail to the right): weekly incomes, house prices, the number of children in a family. A few very large values stretch the tail.
- Negatively skewed (long tail to the left): marks on a very easy test, age at retirement.
In skewed data the mean gets pulled towards the long tail, so the mean and median are not equal. That's a quick test: if the mean and median are very different, the data isn't normal.
For the weekly pay of $500$ workers in a Sydney suburb, the mean is $\$1{,}480$ and the median is $\$1{,}210$. Is this data likely to be normally distributed?
Each "measurement" here is the sum of many small random effects. Press Add 100 a few times and watch the histogram grow. Then turn on Show normal curve.
Switch to Incomes. Why doesn't that one become a bell? Compare where the mean and median land in each case.
Your turn
Which of these is most likely to be normally distributed, and why? (a) The number of goals scored by each player in an A-League season. (b) The mass of $1{,}000$ bags of flour labelled $1$ kg. (c) The price of every house sold in NSW last year.
Show the answer
(b). Flour bags are filled by a machine aiming at $1$ kg, with small random errors either way, so the masses form a symmetric bell around the mean. Goals (a) are positively skewed, since most players score few and a handful score many. House prices (c) are also positively skewed, with a long tail of very expensive homes.
In a normal distribution:
The curve is symmetric with one peak in the middle, so the mean, median and mode all sit at the centre.
Which dataset is most likely to be approximately normal?
Height comes from many small random effects and is symmetric about the mean. Incomes and pets are positively skewed, and a primary school's ages are bunched by year group, not bell-shaped.
Two normal curves have the same mean. Curve A has $s = 3$ and curve B has $s = 8$. Compared with A, curve B is:
A bigger standard deviation spreads the data out. The area must stay the same, so the curve is lower and wider. The centre doesn't move, because the means are equal.
A dataset has mean $54$ and median $61$. It is most likely:
The mean is less than the median, so something is pulling it down: a long tail to the left. That is negatively skewed.
The total area under a normal curve represents:
The whole area under the curve stands for all of the data, which is $100\%$.
A normal distribution of test marks has mean $65$. What is the median?
In a normal distribution the mean and median are equal, so the median is $65$.
Which two values completely describe the position and shape of a normal curve?
The mean sets where the centre is and the standard deviation sets how spread out the curve is. Those two numbers fix the curve.
A histogram has a peak on the left and a long tail stretching to the right. This data is:
Skew is named after the long tail. A tail to the right (towards the positive end) is positive skew.
Exam style: The weekly hours worked by casual staff at a supermarket have mean $18.2$ and median $14.5$. A manager says the hours are normally distributed. Is the manager correct?
In normal data the mean equals the median. Here $18.2 - 14.5 = 3.7$ hours, a big gap, with the mean pulled up by a few staff working many hours. That is positive skew.
Exam style: A machine fills juice bottles. The volumes are normally distributed with mean $500$ mL. Which statement must be true?
For normal data the median equals the mean, so half the values lie above $500$ mL and half below. The volumes still vary, so the standard deviation isn't $0$.