The Normal Distribution (Edexcel GCSE Statistics)

The Normal Distribution

What is a normal distribution?

• A normal distribution is a probability distribution that can be used with continuous quantities

  • The distribution is symmetrical and bell-shaped

  • The mean, median, and mode are all equal

• The notation for a normal distribution is  

  • is the mean of the distribution

  • is called the variance of the distribution

    • it is a measure of how spread out the data is

    • it is equal to the standard deviation () squared

• If the mean changes but the standard deviation (or variance) stays the same

  • then the shape of the distribution stays the same

• If the standard deviation (or variance) increases but the mean stays the same then the distribution is stretched out horizontally

  • A small standard deviation (small variance) means a tall curve with a narrow centre

  • A large standard deviation (large variance) leads to a short curve with a wide centre

When is a normal distribution a suitable model for a distribution of data?

• To use a normal distribution to model a distribution of data, the following must be true

  • The data must be continuous

    • e.g. height or weight (in the real world, heights and weights of populations are often normally distributed)

  • The distribution must be symmetrical and bell-shaped

    • Most data points near the middle

    • Decreasing evenly to right and left

  • The mean, median and mode must be approximately equal

    • They don't need to be exactly equal, but should be close

    • This means that a normal distribution is not appropriate for skewed data

How are the data points distributed in a normal distribution?

• For a normal distribution

  • The mean (and mode and median) is μ

  • The standard deviation is

    • Remember that standard deviation is the square root of the variance

• Of all the data values:

  • Approximately 68% (just over two thirds) of the data lies within one standard deviation of the mean ()

  • Approximately 95% of the data lies within two standard deviations of the mean ()

  • Nearly all of the data (99.7%) lies within three standard deviations of the mean ()

• Because the distribution is symmetrical, those regions can be divided into equal halves

  • Approximately 34% (just over one third) of the data lies

    • between and

    • or between and

  • Approximately 47.5% of the data lies

    • between and

    • or between and

  • Nearly half the data (49.9%) lies

    • between and

    • or between and

• Note that values more than 3 standard deviations from the mean are very unusual

  • Such extreme values are considered to be outliers