Transcript Powerpoint (Measures of Central Tendency

Chapter 3.4
Measures
of
Central Tendency
Measures
of Central Tendency
include mean, median and
mode
The
mean is the sum of values
in a set divided by the number
of values in the set of data.
The
median is the middle
value when the data is ordered
from least to greatest.
The
mode is the number that
occurs most often in the set of
data.
Example 1
 Example:
Suppose a class
received the following test
scores out of 100:
 61, 76, 89, 72, 65, 71, 61, 83,
45, 68, 62, 59, 71, 68, 69, 86
a)
 What
is the mean?
 Mean
=
Total
of
scores
___________________________
Number of scores
 Mean
=
Total
of
scores
___________________________
Number of scores
=
1106
_______
16
 Mean
=
Total of scores
Number of scores
=
1106
_______
16
=
69.125
___________________________
b)
 What
is the median?
b)
 What
is the median?
 First, order the data from least
to greatest
b)
 What
is the median?
 First, order the data from least
to greatest
 45, 59, 61, 61, 62, 65, 68, 68,
69, 71, 71, 72, 76, 83, 86, 89
 Then,
figure out what the
middle value is or what the
middle values are.
 If there is only one middle
value, then that number is
the median.
 If
there are two middle values,
then you determine the
median by adding those values
and dividing by 2.
 Going
back to our numbers,
placed in order:
 45, 59, 61, 61, 62, 65, 68, 68,
69, 71, 71, 72, 76, 83, 86, 89
 The
middle numbers are 68
and 69, so our median is
 The
middle numbers are 68
and 69, so our median is
 Median = _____________
68 + 69
2
 The
middle numbers are 68
and 69, so our median is
 Median = 68
+ 69
_____________
2
= 68.5
c)
 What
is the mode?
c)
 What
is the mode?
 The mode is the number which
occurs most often. Since 61,
68 and 71 each occur twice,
the three modes are 61, 68
and 71.
Example 2
Suppose a charity fundraiser
gave out the following prizes:
Example 2
Suppose a charity fundraiser
gave out the following prizes:
One $5000 prize, four $1000
prizes, eight $500 prizes, and
eighty $10 prizes.
a)
What is the mean?
A total of 93 prizes were given
out.
A total of 93 prizes were given
out.
The total amount of money
given out is calculated as
follows …
$5000 x 1 =
$5000
$5000 x 1 =
$1000 x 4 =
$5000
$4000
$5000 x 1 =
$1000 x 4 =
$500 x 8 =
$5000
$4000
$4000
$5000 x 1
$1000 x 4
$500 x 8
$10 x 80
=
=
=
=
$5000
$4000
$4000
$800
$5000 x 1
$1000 x 4
$500 x 8
$10 x 80
Total
= $5000
= $4000
= $4000
=
$800
= $13800
Mean =
Total Money Given Out
Number of Prizes Given
____________________________________
Mean =
=
Total
Money Given Out
___________________________________
Number of Prizes Given
$13800
__________
93
Mean =
=
=
Total Money Given Out
Number of Prizes Given
$13800
__________
93
$148.39
____________________________________
b)
What is the median?
There are 93 prizes, 80 of
which have a value of $10.
Therefore, the median prize
value is $10.
c)
What is the mode?
The number which occurs
most often is 10, so the mode
is 10.
Chapter 3.5
Measures of
Spread
Measures
of Spread
include range and standard
deviation.
The
range is the difference
between the greatest and
least values in a set of
data.
The
standard deviation is
the typical distance of a
particular value from the
mean. The greater the
standard deviation, the
greater the spread of the
data.
The
formula for standard
deviation is quite
complicated.
Standard
deviation =
Standard
deviation =
_________________________________________________
√
(x1-mean)2 + (x2-mean)2 + … + (xn-mean)2
________________________________________________
n
Example
 The Pittsburgh Penguins
have recorded the following
point totals in the past five
years
Example
 The Pittsburgh Penguins
have recorded the following
point totals in the past five
years
 58, 105, 102, 99, 101
 What is the range of points,
and what is the standard
deviation?
 The highest point total is 105
and the lowest point total is
58. The range is 47
 To determine the standard
deviation, we first need to
determine the mean.
 Mean =_______________________________________
58 + 105 + 102 + 99 + 101
5
 Mean =_______________________________________
58 + 105 + 102 + 99 + 101
5
=
465
5
_______
 Mean =_______________________________________
58 + 105 + 102 + 99 + 101
5
=
=
465
5
_______
93
 Refer to table in your notes to
calculate standard deviation.