The document discusses key aspects of the normal distribution. It explains that a normal distribution is symmetrical and that percentages of observations can be determined within standard deviations of the mean. It also outlines the 68-95-99.7% rule where 68% of values lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. The document provides details on how to calculate standard scores or z-scores.

1. Further Mathematics
Normal Distribution
K McMullen 2012

2. Normal Distributions
Please note that you need the corresponding
graphs to understand what is written in the
powerpoint
Also, the standard score equation has not been
given in the last slide
K McMullen 2012

3. Normal Distribution
The normal distribution is used when the data set
is roughly symmetrical
In normal distributions, the percentage of
observations that lie within a certain number of
standard deviations of the mean can always be
determined.
Generally, we focus on only one, two or three
standard deviations from the mean
K McMullen 2012

4. Normal Distribution
The normal distribution always has the mean
directly in the middle
The standard deviations are evenly spaced from
the mean
99.97% of the data falls within 3 standard
deviations of the mean therefore, when drawing
the standard deviation, we tend to only go 3
standard deviations below the mean and 3
standard deviations above the mean
K McMullen 2012

5. Normal Distribution
Looking within the standard deviations
The 68-95-99.7% rule
68% of the values lie within 1 standard deviation
of the mean
95% of the values lie within 2 standard deviations
of the mean
99.7% of the values lie within 3 standard
deviations of the mean
K McMullen 2012

6. Normal Distribution
Looking above or below the standard deviations
16% is above 1 standard deviation from the mean
16% is below 1 standard deviation from the mean
2.5% is above 2 standard deviations from the
mean
2.5% is below 2 standard deviations from the
mean
0.15% is above 3 standard deviations from the
mean
0.15% is below 3 standard deviations from the
mean
K McMullen 2012

7. Normal Distribution
Standard Scores (z-scores)
The 68-95-99.97% rule makes the standard
deviation a natural measuring stick for normally
distributed data
Standardising data allows us to see how a value
relates to the mean
The mean is 0, therefore:
A positive z-score indicates the data value lies
above the mean
A zero z-score indicates the data value is equal to
the mean
A negative z-score indicates that the data value lies
below the mean
K McMullen 2012

8. Normal Distribution
Calculating a z-score:
K McMullen 2012