Transcript Normal Distribution

Shapes of Distributions
The Normal Distribution
The three histograms shown below represent the marks scored by
students sitting 3 different tests.
A
Left-Skewed
B
C
Symmetrical
Right-Skewed
A symmetrical distribution occurs
often and is known as the
NORMAL distribution. It is
characterized by the bell-shaped
curve that can be drawn through
the top of each bar.
Which of the distribution below are
approximately normal, right skewed, left skewed, or neither?
A
B
C
Right Skewed
D
E
F
Neither
Normal
G
H
Neither
Neither
Normal
Left Skewed
I
Right Skewed
Normal
The Normal Distribution
In a normal distribution most of the data is grouped about the mean and the
rest of the data tails off symmetrically on either side. Approximately 68%
of the data lies within one standard deviation of the mean and 95% of the
data within two standard deviations
of the mean.
The Normal Distribution
In a normal distribution most of the data is grouped about the mean and the
rest of the data tails off symmetrically on either side. Approximately 68%
of the data lies within one standard deviation of the mean and 95% of the
data within two standard deviations
of the mean.
 68%
x -1σ
x
x
x +1σ
The Normal Distribution
In a normal distribution most of the data is grouped about the mean and the
rest of the data tails off symmetrically on either side. Approximately 68%
of the data lies within one standard deviation of the mean and 95% of the
data within two standard deviations
of the mean.
 68%
x -1σ
x
x +1σ
 95%
x -2σ
x
x +2σ
The Normal Distribution
As an example consider a test in which the results are normally distributed.
If the mean is 20 and the standard deviation is 3 then roughly:
are between
17 and 23.
68% of
the marks
11
14
17
20
23
26
are between
14 and 26.
95% of
the marks
11
29
14
17
20
23
26
29
The Normal Distribution
Example 1
The results of a test are normally
distributed with a mean score of 56
and a standard deviation of 4.
(a) What percentage of students should
score between 52 and 60 marks?
 68%
Ans: 68%
(b) What percentage of students should
score between 48 and 64 marks?
Ans: 95%
(c) What percentage of students should
score over 64 .
Ans: 2.5%
x -1σ
x
x +1σ
Questions
 95%
x-2σ
x
x +2σ
The Normal Distribution
Example 2
3000 bars of chocolate have a mean
weight of 120 g and a s.d. of 3 g.
Assuming that they are normally
distributed:
(a) How many bars would you expect to
weigh
between 117 g and 123 g?
Ans:
2040
(b) How many bars would you expect to
weigh
between 114 g and 126 g?
Ans:
2850
(c) How many bars would you expect to
weigh less than 114 g?
Ans: 75
 68%
x -1σ
x
x +1σ
 95%
x-2σ
x
x +2σ
The Normal Distribution
Example 3
In a sample of 2000 adult males, the
mean height was found to be 175 cm
and the s.d. was 5 cm. Assuming a
normal distribution:
(a) How many adults would you expect to
be between 170 cm and 180 cm?
 68%
x -1σ
x
x +1σ
Ans: 1360
(b) Find the heights between which the
central 95% of the distribution
should lie.
 95%
Ans: 165 – 185 cm
(c) How many adults would you expect to
be taller than 185 cm?
Ans: 50
x-2σ
x
x +2σ
The Normal Distribution
Example 4
The time taken for Jenny to walk to
school is normally distributed with a
mean time of 18 minutes and a s.d. of
4 minutes. Calculate:
(a) The number of days last academic
year (192 days) on which you would
expect Jenny to take between 10
and 26 minutes to arrive at school.
Ans: 182
(b) The number of days on which you
would expect her to take less than
14 minutes.
 68%
x -1σ
x
x +1σ
 95%
Ans: 31
x-2σ
x
x +2σ
The Normal Distribution
Example 5
The mean lifetime of a particular type
of battery is 8½ hours with a s.d. of
30 minutes. In a batch of 5 000
batteries and assuming the lifetimes
are normally distributed :
(a) How many of the batteries would you
expect to last for between 8 and 9
hours?
Ans: 3400
(b) How many of the batteries would you
expect to last for between 7½ and
9½ hours?
 68%
x -1σ
x
x +1σ
 95%
Ans: 4750
x-2σ
x
x +2σ
Let’s add some more information…
Example Question 1
Worksheet 1
The results of a test are normally distributed with a mean score of 56 and a standard deviation of 4. Find:
(a) What percentage of students score between 52 and 60 marks?
(b) What percentage of students score between 48 and 64 marks?
(c) What percentage of students score over 64 .
Example Question 2
3000 bars of chocolate have a weight of 120 g and a s.d. of 3 g. Assuming that they are normally distributed:
(a) How many bars would you expect to weigh between 117 g and 123 g?
(b) How many bars would you expect to weigh between 114 g and 126 g?
(c) How many bars would you expect to weigh less than 114 g?
Question 1
Worksheet 2
In a sample of 2000 adult males, the mean height was found to be 175 cm and the s.d. was 5 cm. Assuming a normal
distribution:
(a) How many adults would you expect to be between 170 cm and 180 cm?
(b) Find the heights between which the central 95% of the distribution should lie.
(c) How many adults would you expect to be taller than 185 cm?
Question 2
The time taken for Jenny to walk to school is normally distributed with a mean time of 18 minutes and a s.d. of 4
minutes. Calculate:
(a) The number of days last academic year (192 days) on which you would expect Jenny to take between 10 and 26
minutes to arrive at school.
(b) The number of days on which you would expect her to take less than 14 minutes.
Question 3
The mean lifetime of a particular type of battery is 8½ hours with a s.d. of 30 minutes. In a batch of 5 000 batteries
and assuming the lifetimes are normally distributed :
(a) How many of the batteries would you expect to last for between 8 and 9 hours?
(b) How many of the batteries would you expect to last for between 7½ and 9½ hours?