More about the normal distribution
We start this section, Z-scores and stanines, with some extra information about normal distributions. Remember that a normal curve has the same properties as a histogram. Indeed, a normal distribution can be treated as an extremely large 'population' of values that might underly the data that we have collected.
As a result, a normal distribution has a mean, median, standard deviation, and interquartile range that are defined in a similar way to these summaries of a data set. For example, the median of a normal distribution splits its area into two. Since the distribution is symmetric, the distribution's median is equal to its mean. The quartiles similarly split the area under the normal curve into four equal parts.
Normal mean and standard deviation
The mean and standard deviation of the normal distribution are its most important characteristics because the two parameters of the normal distribution, µ and σ, and are equal to its mean and standard deviation.
If a normal distribution is fitted to a data set, the best fit to the data is obtained by setting the two parameters to the corresponding values from the data — the sample mean and sample standard deviation.
The diagram below shows a histogram of marks (out of 60) for 60 year 7 students in a vocabulary test, with a superimposed normal probability density function.
Use the sliders to adjust the normal parameters to obtain as close as possible a match to the histogram.
Click the button Best fit to set the normal parameters to the mean and sample standard deviation of the data. These give the best fit to the data.