Normal distributions (AQA GCSE Statistics): Revision Notes
Normal distributions
When to use a normal distribution
A normal distribution serves as an appropriate mathematical model when your data meets these three key conditions:
- Continuous data: The data can take any value within a range (like height, weight, or time)
- Symmetrical and bell-shaped: When you plot the data, it forms a smooth, symmetrical curve that looks like a bell
- Central tendency: The mode, median, and mean are all approximately the same value
When these conditions are met, you can use the powerful properties of normal distributions to solve probability problems and make predictions about your data.
The normal distribution is one of the most important probability distributions in statistics because it appears naturally in many real-world phenomena, from measurement errors to biological characteristics.
How values are distributed
Normal distributions follow a very predictable pattern known as the empirical rule. This tells us exactly what percentage of values fall within certain distances from the mean (μ):
- 68% of all values lie within 1 standard deviation (±1σ) of the mean
- 95% of all values lie within 2 standard deviations (±2σ) of the mean
- 99.7% of all values lie within 3 standard deviations (±3σ) of the mean
This creates a bell-shaped curve where most values cluster around the mean, with fewer values appearing as you move further away from the centre.
The empirical rule is sometimes called the "68-95-99.7 rule" and is fundamental to understanding how normal distributions behave. This pattern holds true for ANY normal distribution, regardless of its specific mean and standard deviation values.
Calculating standard deviations from the mean
To work out how many standard deviations a particular value is from the mean, use this essential formula:
This calculation tells you exactly where a value sits on the normal distribution curve, which is crucial for finding probabilities.
This formula is the key to converting any normal distribution problem into a standard normal distribution problem. Master this formula and you'll be able to tackle any normal distribution question!
Worked example: Daffodil heights
Worked Example: Finding Mean and Standard Deviation
Question: The heights of a species of daffodil are normally distributed. 2.5% of the heights are greater than 16.5cm, and 50% of the heights are greater than 13.5cm.
(a) Find the mean and standard deviation
Since 50% of heights are greater than 13.5cm, this value represents the mean (because normal distributions are symmetrical around the mean).
Therefore: μ = 13.5cm
For the standard deviation, we know that 2.5% of heights are greater than 16.5cm. Since the distribution is symmetrical, 50% lies above the mean, so 2.5% represents the extreme upper tail.
In a normal distribution, 95% of values lie within μ ± 2σ, meaning 2.5% lie above μ + 2σ.
So: μ + 2σ = 16.5cm 13.5 + 2σ = 16.5 2σ = 3.0 σ = 1.5cm
(b) Find the probability that heights are greater than 18cm
First, calculate how many standard deviations 18cm is from the mean:
Since 99.7% of values lie within 3 standard deviations of the mean, only 0.3% lie outside this range. Due to symmetry, half of this (0.15%) lies above μ + 3σ.
Therefore: Probability = 0.15% or 0.0015
Key exam tips
Understanding normal distributions requires both conceptual knowledge and practical problem-solving skills. Here are the most important strategies for exam success:
Drawing normal distribution sketches: Always sketch your normal curve showing 3 standard deviations either side of the mean. This helps visualise the problem and spot where your target values fall.
Common trap: Remember that "greater than" means you want the area to the right of your value on the curve, while "less than" means the area to the left.
Symmetry is key: Use the symmetrical property to your advantage. If you know information about one side of the distribution, you can apply it to the other side.
Check your arithmetic: Standard deviation calculations often involve several steps. Double-check each calculation, especially when working backwards from percentages to find μ and σ.
Many students lose marks by rushing through the arithmetic in normal distribution problems. Take time to carefully work through each step, especially when converting between percentages and standard deviations.
Remember!
Key Points to Remember:
- Normal distributions are appropriate for continuous, symmetrical, bell-shaped data where the mean, median, and mode are approximately equal
- The 68-95-99.7 rule tells us exactly how values are spread in any normal distribution
- Use the formula to find how many standard deviations any value is from the mean
- Always sketch the distribution to visualise probability problems
- The symmetrical property of normal distributions is your best friend for solving complex probability questions