Chapter 3.4Chapter 3.4

Measures of Measures of Central Central TendencyTendency

Measures of Central Measures of Central Tendency include mean, Tendency include mean, median and modemedian and mode

The mean is the sum of The mean is the sum of values in a set divided by values in a set divided by the number of values in the number of values in the set of data.the set of data.

The median is the middle The median is the middle value when the data is value when the data is ordered from least to ordered from least to greatest.greatest.

The mode is the number The mode is the number that occurs most often in that occurs most often in the set of data.the set of data.

Example 1Example 1

Example: Suppose a class Example: Suppose a class received the following test received the following test scores out of 100:scores out of 100:

61, 76, 89, 72, 65, 71, 61, 61, 76, 89, 72, 65, 71, 61, 83, 45, 68, 62, 59, 71, 68, 83, 45, 68, 62, 59, 71, 68, 69, 86 69, 86

a)a)

What is the mean?What is the mean?

Mean = Total of scoresMean = Total of scores Number of scoresNumber of scores

___________________________

Mean = Total of scoresMean = Total of scores Number of scoresNumber of scores

= 1106= 1106 1616

___________________________

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Mean = Total of scoresMean = Total of scores Number of scoresNumber of scores

= 1106= 1106 1616

= 69.125= 69.125

___________________________

_______

b)b)

What is the median?What is the median?

b)b)

What is the median?What is the median? First, order the data from First, order the data from

least to greatestleast to greatest

b)b)

What is the median?What is the median? First, order the data from First, order the data from

least to greatestleast to greatest 45, 59, 61, 61, 62, 65, 68, 45, 59, 61, 61, 62, 65, 68,

68, 69, 71, 71, 72, 76, 83, 68, 69, 71, 71, 72, 76, 83, 86, 8986, 89

Then, figure out what the Then, figure out what the middle value is or what middle value is or what the middle values are.the middle values are.

If there is only one middle If there is only one middle value, then that number is value, then that number is the median.the median.

If there are two middle If there are two middle values, then you values, then you determine the median by determine the median by adding those values and adding those values and dividing by 2. dividing by 2.

Going back to our Going back to our numbers, placed in order:numbers, placed in order:

45, 59, 61, 61, 62, 65, 68, 45, 59, 61, 61, 62, 65, 68, 68, 69, 71, 71, 72, 76, 83, 68, 69, 71, 71, 72, 76, 83, 86, 8986, 89

The middle numbers are The middle numbers are 68 and 69, so our median 68 and 69, so our median is is

The middle numbers are 68 The middle numbers are 68 and 69, so our median is and 69, so our median is

Median = 68 + 69Median = 68 + 69 22

_____________

The middle numbers are 68 The middle numbers are 68 and 69, so our median is and 69, so our median is

Median = 68 + 69Median = 68 + 69 22 = 68.5= 68.5

_____________

c)c)

What is the mode?What is the mode?

c)c)

What is the mode?What is the mode? The mode is the number The mode is the number

which occurs most often. which occurs most often. Since 61, 68 and 71 each Since 61, 68 and 71 each occur twice, the three occur twice, the three modes are 61, 68 and 71.modes are 61, 68 and 71.

Example 2Example 2

Suppose a charity fundraiser Suppose a charity fundraiser gave out the following prizes:gave out the following prizes:

Example 2Example 2

Suppose a charity fundraiser Suppose a charity fundraiser gave out the following prizes:gave out the following prizes:One $5000 prize, four $1000 One $5000 prize, four $1000 prizes, eight $500 prizes, and prizes, eight $500 prizes, and eighty $10 prizes.eighty $10 prizes.

a)a)

What is the mean?What is the mean?

A total of 93 prizes were given A total of 93 prizes were given out.out.

A total of 93 prizes were given A total of 93 prizes were given out.out.The total amount of money The total amount of money given out is calculated as given out is calculated as follows …follows …

$5000 x 1 = $5000$5000 x 1 = $5000

$5000 x 1 = $5000$5000 x 1 = $5000$1000 x 4 = $4000$1000 x 4 = $4000

$5000 x 1 = $5000$5000 x 1 = $5000$1000 x 4 = $4000$1000 x 4 = $4000 $500 x 8 = $4000$500 x 8 = $4000

$5000 x 1 = $5000$5000 x 1 = $5000$1000 x 4 = $4000$1000 x 4 = $4000 $500 x 8 = $4000$500 x 8 = $4000 $10 x 80 = $800$10 x 80 = $800

$5000 x 1 = $5000$5000 x 1 = $5000$1000 x 4 = $4000$1000 x 4 = $4000 $500 x 8 = $4000$500 x 8 = $4000 $10 x 80 = $800$10 x 80 = $800 Total = $13800Total = $13800

Mean = Total Money Given OutMean = Total Money Given Out Number of Prizes Number of Prizes

GivenGiven

____________________________________

Mean = Total Money Given OutMean = Total Money Given Out Number of Prizes Given Number of Prizes Given == $13800 $138009393

___________________________________

__________

Mean = Total Money Given OutMean = Total Money Given Out Number of Prizes Given Number of Prizes Given == $13800 $138009393== $148.39 $148.39

____________________________________

__________

b)b)

What is the median?What is the median?

There are 93 prizes, 80 of There are 93 prizes, 80 of which have a value of $10. which have a value of $10. Therefore, the median prize Therefore, the median prize value is $10.value is $10.

c)c)

What is the mode?What is the mode?

The number which occurs The number which occurs most often is 10, so the mode most often is 10, so the mode is 10.is 10.

Chapter 3.5Chapter 3.5

Measures of Measures of SpreadSpread

Measures of Spread Measures of Spread include range and include range and standard deviation.standard deviation.

The range is the difference The range is the difference between the greatest and between the greatest and least values in a set of least values in a set of data.data.

The standard deviation is The standard deviation is the typical distance of a the typical distance of a particular value from the particular value from the mean. The greater the mean. The greater the standard deviation, the standard deviation, the greater the spread of the greater the spread of the data.data.

The formula for standard The formula for standard deviation is quite deviation is quite complicated. complicated.

Standard deviation = Standard deviation =

Standard deviation = Standard deviation =

((xx11-mean)-mean)22 + ( + (xx22-mean)-mean)22 + … + ( + … + (xxnn-mean)-mean)22 nn√_________________________________________________

________________________________________________

ExampleExample

The Pittsburgh Penguins The Pittsburgh Penguins have recorded the following have recorded the following point totals in the past five point totals in the past five yearsyears

ExampleExample

The Pittsburgh Penguins The Pittsburgh Penguins have recorded the following have recorded the following point totals in the past five point totals in the past five yearsyears

58, 105, 102, 99, 10158, 105, 102, 99, 101

What is the range of points, What is the range of points, and what is the standard and what is the standard deviation?deviation?

The highest point total is 105 The highest point total is 105 and the lowest point total is and the lowest point total is 58. The range is 4758. The range is 47

To determine the standard To determine the standard deviation, we first need to deviation, we first need to determine the mean.determine the mean.

Mean = Mean = 58 + 105 + 102 + 99 + 10158 + 105 + 102 + 99 + 10155

_______________________________________

Mean = Mean = 58 + 105 + 102 + 99 + 10158 + 105 + 102 + 99 + 10155

== 465465 55

_______________________________________

_______

Mean = Mean = 58 + 105 + 102 + 99 + 10158 + 105 + 102 + 99 + 10155

== 465465 55

= 93= 93

_______________________________________

_______

Refer to table in your notes to Refer to table in your notes to calculate standard deviation.calculate standard deviation.