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Further Mathematics
Normal Distribution



             K McMullen 2012
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
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
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
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
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
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
Normal Distribution
Calculating a z-score:




                         K McMullen 2012

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Further5 normal distribution

  • 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