Normally Distributed Data
Learn about normally distributed data for NSW Year 12 Mathematics Standard 2. In this topic you recognise the symmetric bell-shaped curve, understand why the mean, median and mode are approximately equal, and read the mean and standard deviation straight off a labelled bell curve.
These are the core properties of the normal distribution — the starting point for the empirical (68–95–99.7) rule and z-scores later in the course. Everyday quantities like heights, masses, test scores and temperatures are often normally distributed, making this a key Standard 2 skill for interpreting real-world data.
Theory
Normally distributed data forms a symmetric bell-shaped curve. This Year 12 Standard 2 (NSW) guide shows how to recognise the bell shape, why the mean, median and mode are all equal and sit at the centre, and how to read the mean and standard deviation straight off a bell curve.
Data is normally distributed when its graph is a smooth, symmetric bell-shaped curve. Most values cluster near the centre and the frequency tails off evenly on both sides, giving the curve a single central peak.
Because the curve is symmetric, the mean, median and mode are all approximately equal and sit together at the centre, directly below the peak: \(\text{mean} \approx \text{median} \approx \text{mode}\).
The mean is read as the value at the centre of the bell. The standard deviation is read as the distance between two adjacent one-SD marks (the marks spaced evenly either side of the mean). Exactly half — 50% — of the data lies below the mean. This is the foundation for the empirical rule and z-scores in Year 12 Standard 2 (NSW).
For normally distributed data the three measures of centre coincide:
The vertical marks on a bell curve are one standard deviation apart, so the standard deviation \(\sigma\) is the gap between the mean \(\mu\) and the next mark:
How to read a normal (bell) curve
- Check the shape. Confirm the graph is a single, symmetric bell — one peak, mirror-image sides.
- Read the mean. The mean is the value on the horizontal axis at the centre, directly below the peak.
- Median and mode. For normal data these equal the mean, so they sit at the centre too.
- Read the standard deviation. The marks are one SD apart, so the SD is the distance between the mean and the next mark.
For symmetric (normal) data the three measures of centre should coincide.
| \(4+6+7+7+7+8+10\) | \(=\) | \(49\) |
| \(\text{mean}=\dfrac{49}{7}\) | \(=\) | \(7\) |
| \(\text{median}\) | \(=\) | \(7\) |
| \(\text{mode}\) | \(=\) | \(7\) |
All three equal 7 and sit at the centre — the data is symmetric, consistent with a normal distribution.
The mean is the value at the centre of the bell, directly below the peak.
| \(\text{peak sits above}\) | \(\) | \(\text{the centre}\) |
| \(\text{central label}\) | \(=\) | \(180\) |
| \(\therefore\ \text{mean}\) | \(=\) | \(180\text{ g}\) |
The SD is the gap between the mean and the next mark.
| \(\text{mean (centre)}\) | \(=\) | \(22\) |
| \(\text{next mark}\) | \(=\) | \(25\) |
| \(\text{SD}\) | \(=\) | \(25-22\) |
| \(\) | \(=\) | \(3\) |
The standard deviation is 3 \(^{\circ}\)C.
A normal curve is symmetric about the mean.
| \(\text{median}\) | \(\approx\) | \(\text{mean}=40\text{ mm}\) |
| \(\text{below the mean}\) | \(=\) | \(\tfrac{1}{2}\) |
| \(\) | \(=\) | \(50\%\) |
The median is 40 mm and 50% of nails are shorter than 40 mm.
Common pitfalls
Frequently asked questions
What does normally distributed data look like?
It forms a smooth, symmetric bell-shaped curve when graphed. Values cluster around a central value and thin out evenly on both sides, giving a single peak in the middle.
How do you find the mean from a bell curve?
The mean is the value on the horizontal axis at the centre of the bell, directly below the peak. Read straight down from the highest point of the curve to the axis.
Are the mean, median and mode the same in a normal distribution?
Yes, approximately. Because a normal curve is symmetric, the mean, median and mode all sit at the centre and are approximately equal. If they differ a lot, the data is not normal.
How do you read the standard deviation off a bell curve?
The vertical marks either side of the mean are spaced one standard deviation apart, so the standard deviation is the distance between the mean and the next mark, for example 65 minus 60 equals 5.
What percentage of data is below the mean in a normal distribution?
Exactly 50 percent. The curve is symmetric about the mean, so the mean splits the data into two equal halves, with 50 percent below the mean and 50 percent above it.
Which kinds of data are usually normally distributed?
Naturally varying measurements such as people's heights, the masses of fruit, test scores, daily temperatures and manufactured part sizes are often normally distributed, because most values sit near an average with fewer extreme values.