1. “Teach A Level Maths” Statistics 1
The Normal Distribution
© Christine Crisp

2. Statistics 1 AQA Edexcel
Normal Distribution diagrams in the examples and exercises in this presentation have been drawn using FX Draw ( available from Efofex at )
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3. Suppose we have a crate of apples which are to be sorted by weight into small, medium and large. If we wanted 25% to be in the large category, we would need to know the lowest weight a “large” apple could be.
( Here I am using weight in the everyday sense; the quantity measured in kilograms and grams. If you are a physicist you will refer to mass. )
To solve a problem like this we can use a statistical model.
A model often used for continuous quantities such as weight, volume, length and time is the Normal Distribution.
The Normal Distribution is an example of a probability model.

4. Characteristics of the Normal Distribution
If we were to show the weights of a large number of our apples in a histogram we might get this:
There are not many very light . . .
or very heavy apples.
The distribution is fairly symmetric.

5. Characteristics of the Normal Distribution
The Normal distribution model is a symmetric bell-shaped curve. We fit it as closely as possible to the data.
The Normal Distribution curve
To fit the curve we use the mean, m, and variance, , of the data. These are the parameters of the model.
If X is the random variable “ the weight of apples”, we write
Reminder: s is the standard deviation.

6. Characteristics of the Normal Distribution
The Normal distribution model is a symmetric bell-shaped curve. We fit it as closely as possible to the data.
The Normal Distribution curve
For this curve we might have
If a question gives me the standard deviation, I often write the variance in this form instead of simplifying.

7. The axis of symmetry of the Normal distribution passes through the mean.
e.g.

8. A smaller variance “squashes” the distribution closer to the mean.
e.g.

9. P x (X = ) Finding probabilities
When we had a discrete distribution we could find a probability by using a formula.
e.g. The r.v. X has probability distribution function (p.d.f.) given by P x
(X = )
For a continuous distribution, a probability is given by an area under the graph of the p.d.f.

10. For example, the probability that an apple taken at random weighs less than 200 grams is given by the area to the left of a line through 200.
The total area gives the sum of the probabilities so equals 1.
The p.d.f. of the Normal curve is
Since this is a difficult function the probabilities have been worked out and listed in a table.

11. Finding Probabilities
Before we do an example, find the table of probabilities in your formulae book.
The table gives probabilities for the random variable Z where
Notice the diagram at the top of the page. The shading shows that the probabilities given are always less than the z value used.
( I will usually write the values to 4 d.p. )
Since we may need to use these values to find others, we will always draw a sketch.

12. e.g.1 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(a)
(use the table )
(b)
is a lot to write so we can write f(1·6) which means the area to the left of 1·6.

13. e.g.1 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(a)
(use the table )
(b)
Tip: It’s useful to always check
the answer: an area less than 0·5
corresponds to less than half the area and vice versa.

14. e.g.1 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(c)

15. e.g.1 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(c)
(use the table )

16. Special Cases
The probability of any single value is zero.
e.g.
( There is no area. )
Also, using is the same as using <
e.g.
Neither property holds for discrete distributions.

17. SUMMARY
The Normal Distribution is continuous and symmetric about the mean.
An area under the curve gives a probability.
The formula booklet gives a table of probabilities for the random variable Z where
The table gives values of This is written as z
When using the table we always draw a sketch showing the required probability.

18. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Since we want ( the standard deviation ), the variance, , is also equal to 1.
The mean can be any value, so let
We can find the percentage from the probability, so we need the probability that Z is between – 1 and + 1.
The easiest way to find this area is to find
and subtract

19. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Since we want ( the standard deviation ), the variance, , is also equal to 1.
The mean can be any value, so let
We can find the percentage from the probability, so we need the probability that Z is between – 1 and + 1.
The easiest way to find this area is to find
and subtract
then multiply by 2.

20. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Since we want ( the standard deviation ), the variance, , is also equal to 1.
The mean can be any value, so let
We can find the percentage from the probability, so we need the probability that Z is between – 1 and + 1.
Can you see what equals without using the table?

21. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Since we want ( the standard deviation ), the variance, , is also equal to 1.
The mean can be any value, so let
We can find the percentage from the probability, so we need the probability that Z is between – 1 and + 1.

22. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Since we want ( the standard deviation ), the variance, , is also equal to 1.
The mean can be any value, so let
We can find the percentage from the probability, so we need the probability that Z is between – 1 and + 1.
The percentage is approximately 68%.

23. Exercise
1. If Z is a random variable with distribution
2. Find the percentage of the Normal distribution that lies within (a) 2 standard deviations either side of the mean and (b) 3 standard deviations either side of the mean.
find (a) (b) (c)

24. Solutions:
1(a)
1(b)

25. Solutions:
1(c)

26. 2. Find the percentage of the Normal distribution that lies within (a) 2 standard deviations either side of the mean and (b) 3 standard deviations either side of the mean.
Solution: Let
(a) We want
Approximately 95% of the Normal distribution lies within 2 standard deviations of the mean.
(b) The method is the same. The answer is approx. 99·8%.

27. 68% 1 s.d. : 68% 2 s.d. : 95% 3 s.d. : 99·8% 95% 99·8% SUMMARY
The percentages of the Normal Distribution lying within the given number of standard deviations either side of the mean are approximately:
68%
1 s.d. : 68%
2 s.d. : 95%
3 s.d. : 99·8%
95%
99·8%

28. e.g.3 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(a)
The table only gives probabilities for positive z values so we have to find an equal area that is in the table.

29. Solution:
(b)
This area equals

30. Solution:
(c)
Tip: Work out first or you could make a sign error.
So,

31. NEVER try to do these questions without at least one diagram.
SUMMARY
To find , the area to the left of a negative number, we use
N.B. The procedure for this and all other areas involving negative values can be seen from the diagram.
NEVER try to do these questions without at least one diagram.

32. Exercise
1. If Z is a random variable with distribution
find (a) (b)
(c) (d)
There are 2 methods of doing part (d). See if you can spot them both and use the quicker.

33. = Exercise 1. If Z is a random variable with distribution find (a) (b)
(c) (d)
Solution:
(a) =

34. = Exercise 1. If Z is a random variable with distribution find (a) (b)
(c) (d)
Solution:
(b) =

35. Exercise
1. If Z is a random variable with distribution
find (a) (b)
(c) (d)
Solution:
(c)

36. Exercise
1. If Z is a random variable with distribution
find (a) (b)
(c) (d)
(d)
Solution:
Method 1:
Area equals

37. Exercise
1. If Z is a random variable with distribution
find (a) (b)
(c) (d)
Solution:
(d)
Method 2:

39. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

40. SUMMARY
The Normal Distribution is continuous and symmetric about the mean.
The formula booklet gives a table of probabilities for the random variable Z where
An area under the curve gives a probability.
When using the table we always draw a sketch showing the required probability.
The table gives values of This is written as z

41. Special Cases
The probability of any single value is zero.
e.g.
( There is no area )
Also, using is the same as using <
Neither property holds for discrete distributions.

42. e.g.1 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(a)
(use the table )
(b)

43. Solution:
(c)

44. e.g. 2 Find the percentage of the Normal distribution that lies within 1 standard deviation on either side of the mean.
Solution: Let
We can find the percentage from the probability, so we need the probability that Z is between m – 1 and m + 1 where m is the mean and equals zero.
The percentage is approximately 68%.

45. 1 s.d. : 68% 2 s.d. : 95% 3 s.d. : 99·8% 68% 95% 99·8% SUMMARY
The percentages of the Normal Distribution lying within the given number of standard deviations either side of the mean are approximately:
SUMMARY
1 s.d. : 68%
2 s.d. : 95%
3 s.d. : 99·8%
68%
95%
99·8%

46. e.g.3 If Z is a random variable with distribution
find (a) (b) (c)
Solution:
(a)
The table only gives probabilities for positive z values so we have to find an area equal to this that is in the table.

47. Solution:
(b)
This area equals

48. Solution:
(c)
Tip: Work out
first.
So,

49. NEVER try to do these questions without at least one diagram.
SUMMARY
To find , the area to the left of a negative number, we use
N.B. The procedure for this and all other areas involving negative values can be seen from the diagram.
NEVER try to do these questions without at least one diagram.