Download The Normal Distribution

The normal distribution
The curve to the right was symmetrical.
In this topic we will look at a particular type of
symmetrical curve, the normal or bell curve.
Below is an example.
In normal distribution:
•The frequency graph is ‘bell’ shaped
•The mean, median and mode are all equal
The normal curve
The shape varies with different means and standard deviations.
The thinner the curve,
the smaller the standard deviation.
The fatter the curve,
the larger the standard deviation.
Score
These 3 curves have the same mean.
The curves are all the same
height and width,
So they have the same
standard deviation.
Score
These 3 curves have different means.
Area under the normal curve
For any normal distribution:
•100% of the scores lie under the normal curve;
•68% of the scores lie within 1 standard deviation of the mean,
•or 68% lie within ẍ ± σn;
•95% of the scores lie within 2 standard deviations of the mean,
•or 95% lie within ẍ ± 2σn;
•99·7% of the scores lie within 3 standard deviations of the mean,
•or 99·7 % lie within ẍ ± 3σn;
•0·3% of the scores are “in the tail”.
68%
x n
x x 
n
95%
x  2 n
x
99·7%
x  2 n
x  3 n
x
x  3 n
Area and probability
68% within 1SD,
34% each side
95% within 2SD,
47·5% each side,
 47·5  34 = 13·5%
99·7% within 3SD,
49·85% each side,
 49·85  47·5 = 2·35%
100  99·7 = 0·3%
outside 3SD’s
0·15% each side
0·15 %
13·5 %
2·35 %
34%
We say most scores (68%)
lie within 1SD of the mean.
A score will most
(95%)
lie
34% probably
within 2SD’s of the mean.
A score will almost
probably (99 ·7%) lie
within 3SD’s of the
mean.
13·5 %
0·15 %
2·35 %
Example 1
The average mark on a history test was 71%. If the standard
deviation was 6%, within what limits do
a) 68% of the scores lie?
b) 95% of the scores lie?
c) 99·7% of the scores lie?
You may find it useful to draw the curve and write in the values
a) x   n =
 71 ± 6
between 65 and 77
b) x  2 n 
= 71 ± 2 × 6
between 59 and 83
c) x  3 n 
= 71 ± 3 × 6
between 53 and 89
Example 2
The TDK produces 240 minute tapes with a mean time of 243
minutes standard deviation of 3 minutes
a) What percentage of tapes have between 240 and 246 minutes?
b) What is the probability of getting a tape with less than 240 min?
c) If they produced 15 000 tapes yesterday, how many had more
than 249 min of tape?
a) This is 1SD above and below the mean, 68% of the tapes.
b) Less than 240 is less than 1SD below the mean,
50% - 34% = 16%
c) 249 min is 2SD above the mean
2.35% + 0.15%= 2·5%
2·5 ÷ 100 × 15 000 =375 tapes
Today’s work
Exercise 8D pg 248
#1–3
Exercise 8E pg
#1, 3, 5, 6 – 10, 11ab, 12