Measures of Central Tendency. What Are Measures of Central Tendency?

Measures of Central Measures of Central TendencyTendency

What Are Measures of What Are Measures of Central Tendency?Central Tendency?


By Measures of Central Tendency, By Measures of Central Tendency, we are referring to a variety of we are referring to a variety of

measurements of a set of data. measurements of a set of data.


Each of these measurements will Each of these measurements will result in a single number that result in a single number that represents the data set in a represents the data set in a

specific way. specific way.

There are 3 Measures of There are 3 Measures of Central TendencyCentral Tendency

These are:These are:


•MeanMean


•MeanMean•MedianMedian


•MeanMean•MedianMedian•ModeMode

Let’s begin with an Let’s begin with an explanation of Meanexplanation of Mean

MeanMean

The The MeanMean of a set of data is of a set of data is the the averageaverage value of all of value of all of the numbers in the set.the numbers in the set.

MeanMean

For example, let’s consider this set For example, let’s consider this set of numbers:of numbers:

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

MeanMean

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

First, add up all of the numbers in First, add up all of the numbers in the set.the set.

4 + 7 + 9 + 11 + 14 = 454 + 7 + 9 + 11 + 14 = 45

MeanMean

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

Next, count the “number of Next, count the “number of numbers” in the set.numbers” in the set.

There are 5 numbers in this set.There are 5 numbers in this set.

MeanMean{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

Finally, divide the sum of all the numbers in the Finally, divide the sum of all the numbers in the set by the “number of numbers” in the set.set by the “number of numbers” in the set.

4 + 7 + 9 + 11 + 14 = 4 + 7 + 9 + 11 + 14 = 4545

There are There are 55 numbers in this set. numbers in this set.

45 ÷ 5 = 945 ÷ 5 = 9

MeanMean

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

45 ÷ 5 = 945 ÷ 5 = 9

The The meanmean of this set is equal to 9 of this set is equal to 9

MeanMean

Now, try one on your own:Now, try one on your own:

Determine the mean of this set of data:Determine the mean of this set of data:

{12, 15, 22, 25, 36, 40}{12, 15, 22, 25, 36, 40}

MeanMean

{12, 15, 22, 25, 36, 40}{12, 15, 22, 25, 36, 40}

The mean of this set of data is:The mean of this set of data is:

22 24 25 35

MeanMean

CORRECT!CORRECT!

{12, 15, 22, 25, 36, 40}{12, 15, 22, 25, 36, 40}

The mean of this set of data is:The mean of this set of data is: 2525

MeanMean

Sorry, but that’s not quite right. Perhaps Sorry, but that’s not quite right. Perhaps a little review would help.a little review would help.

Next StepNext Step

Now that you have a good understanding Now that you have a good understanding of mean, let’s go on to the next Measure of mean, let’s go on to the next Measure of Central Tendency which is the of Central Tendency which is the

MedianMedian

MedianMedian

Think about driving down the interstate. Think about driving down the interstate. Most of the time, there is an area of grass Most of the time, there is an area of grass between the two strips of highway.between the two strips of highway.

This grassy area is called the This grassy area is called the medianmedian because it is in the because it is in the middlemiddle of the of the interstateinterstate

MedianMedian

In a similar way, the median of a set of In a similar way, the median of a set of numbers is the number that is exactly in numbers is the number that is exactly in the middle of the set.the middle of the set.

Median ExampleMedian Example

Consider again this set of data:Consider again this set of data:

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

Median ExampleMedian Example

{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

The number 9 has two numbers to its left The number 9 has two numbers to its left and two numbers to its right, so it is in and two numbers to its right, so it is in

the middle of this set of data.the middle of this set of data.

Median ExampleMedian Example{4, 7, 9, 11, 14}{4, 7, 9, 11, 14}

Since the number 9 is in the middle of this set of Since the number 9 is in the middle of this set of data, data, 99 is the is the medianmedian..

On Your OwnOn Your Own

Try a problem where you determine the Try a problem where you determine the median of a set of data on your own now.median of a set of data on your own now.

Try This OneTry This One

Determine the median of this set of data:Determine the median of this set of data:

{3, 6, 12, 19, 23, 29, 32}{3, 6, 12, 19, 23, 29, 32}

Try This OneTry This One

{3, 6, 12, 19, 23, 29, 32}{3, 6, 12, 19, 23, 29, 32}

The median of this set of data is:The median of this set of data is:

6 12 19 23

Good Job!Good Job!

Correct!Correct!

The median of this set of data is The median of this set of data is 1919

Now let’s continue …Now let’s continue …

Ascending OrderAscending Order{3, 6, 12, 19, 23, 29, 32}{3, 6, 12, 19, 23, 29, 32}

One thing that you may notice about the One thing that you may notice about the set of data you just worked with is that it set of data you just worked with is that it is already arranged in ascending order. is already arranged in ascending order.

Ascending OrderAscending Order{3, 6, 12, 19, 23, 29, 32}{3, 6, 12, 19, 23, 29, 32}

This means that the numbers start from This means that the numbers start from the smallest. The next one is the next the smallest. The next one is the next

largest, and they continue on this way to largest, and they continue on this way to the largest one.the largest one.

Ascending OrderAscending Order

When calculating mean or mode, the When calculating mean or mode, the order of the data doesn’t matter.order of the data doesn’t matter.

However, when calculating However, when calculating medianmedian, the , the numbers numbers mustmust be in be in ascendingascending orderorder..

Example of Ascending OrderExample of Ascending Order

You’ll notice that the set of data below is You’ll notice that the set of data below is notnot in ascending order. in ascending order.

{14, 2, 47, 32, 12, 53, 29}{14, 2, 47, 32, 12, 53, 29}


To find median, we must place these numbers in To find median, we must place these numbers in ascending order:ascending order:

{14, 2, 47, 32, 12, 53, 29}{14, 2, 47, 32, 12, 53, 29}

equals:equals:

{2, 12, 14, 29, 32, 47, 53}{2, 12, 14, 29, 32, 47, 53}


The middle number of this set, or the The middle number of this set, or the median, is the number median, is the number 2929..

{2, 12, 14, 29, 32, 47, 53}{2, 12, 14, 29, 32, 47, 53}

PracticePractice

Now try another one on your own. Now try another one on your own.

PracticePractice

Determine the median of the following Determine the median of the following set of data:set of data:

{41, 13, 63, 28, 4, 55, 30} {41, 13, 63, 28, 4, 55, 30}

PracticePractice

{41, 13, 63, 28, 4, 55, 30} {41, 13, 63, 28, 4, 55, 30}


28 30 41 55

Good Job!Good Job!

Correct! Correct!


An Even Number of NumbersAn Even Number of Numbers

So far, all of the data sets we have looked So far, all of the data sets we have looked at to determine median have contained at to determine median have contained an odd number of terms, or numbers.an odd number of terms, or numbers.

It is very easy to find the middle number It is very easy to find the middle number in a set if the number of terms is odd.in a set if the number of terms is odd.


Sometimes though, the number of terms Sometimes though, the number of terms in a set is even.in a set is even.

This requires some additional thought to This requires some additional thought to determine the median of such a set.determine the median of such a set.


For example, consider this set of data:For example, consider this set of data:

{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}


{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}

First of all, these numbers are in First of all, these numbers are in ascending order, so we don’t have to ascending order, so we don’t have to

worry about that.worry about that.

However, there is no number directly in However, there is no number directly in the middle of the data set!the middle of the data set!

Finding the Middle NumberFinding the Middle Number

{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}

When there is an even number of terms When there is an even number of terms in a data set, we must find the middle in a data set, we must find the middle

twotwo numbers. numbers.

In this case, the In this case, the twotwo numbers in the numbers in the middle are 8 and 12.middle are 8 and 12.


{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}

To determine the median of this set of To determine the median of this set of data, we now take the two middle data, we now take the two middle

numbers and find the mean of those.numbers and find the mean of those.


{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}

The two middle numbers are 8 and 12.The two middle numbers are 8 and 12.

Find the mean of 8 and 12:Find the mean of 8 and 12:

8 + 12 = 208 + 12 = 20

20 ÷ 2 = 10 20 ÷ 2 = 10


{2, 5, 8, 12, 13, 16}{2, 5, 8, 12, 13, 16}

8 + 12 = 208 + 12 = 20

20 ÷ 2 = 10 20 ÷ 2 = 10

The median of this set of data is The median of this set of data is 1010..

PracticePractice

Now, try one on your own.Now, try one on your own.

PracticePractice

Determine the median of this set of data:Determine the median of this set of data:

{5, 8, 13, 20, 24, 34, 42, 57}{5, 8, 13, 20, 24, 34, 42, 57}

PracticePractice

{5, 8, 13, 20, 24, 34, 42, 57}{5, 8, 13, 20, 24, 34, 42, 57}


20 22 24 28

Good Job!Good Job!

Correct!Correct!


{5, 8, 13, 20, 24, 34, 42, 57}{5, 8, 13, 20, 24, 34, 42, 57}

Moving On …Moving On …

Now that you have a good grasp of Now that you have a good grasp of the concepts of mean and median, the concepts of mean and median, let’s learn about the mode of a set.let’s learn about the mode of a set.

ModeMode

The mode of a set is the number The mode of a set is the number that appears most often in a set of that appears most often in a set of

data.data.

ModeMode

The word The word modemode begins with the begins with the letters letters mm and and oo … …

The word The word mostmost also begins with the also begins with the letters letters mm and and oo

MoMode = de = MoMostst

Example of ModeExample of Mode{16, 9, 2, 9, 8, 11, 9, 13, 11}{16, 9, 2, 9, 8, 11, 9, 13, 11}

In the data set above, you will notice that the number 9 appears In the data set above, you will notice that the number 9 appears more often than any other number.more often than any other number.

9 appears 9 appears momost often.st often.

9 is the 9 is the momode of this set.de of this set.

(You may also notice that the order of terms did not matter)(You may also notice that the order of terms did not matter)

Practice Finding the ModePractice Finding the Mode

Now, try one on your own.Now, try one on your own.


Determine the mode of the following set Determine the mode of the following set of data:of data:

{31, 64, 89, 64, 31, 102, 31, 29}{31, 64, 89, 64, 31, 102, 31, 29}


{31, 64, 89, 64, 56, 31, 102, 31, 29,}{31, 64, 89, 64, 56, 31, 102, 31, 29,}

The mode of the set of data isThe mode of the set of data is

31 56 64 102

Good Job!Good Job!

Correct!Correct!

{31, 64, 89, 64, 56, 31, 102, 31, 29,}{31, 64, 89, 64, 56, 31, 102, 31, 29,}

The mode of this set of data is The mode of this set of data is 3131

Congratulations!Congratulations!

You have mastered the 3 Measures of You have mastered the 3 Measures of Central Tendency:Central Tendency:

MeanMean, , MedianMedian, and , and ModeMode..

Not Quite!Not Quite!

Sorry, but that’s not quite right.Sorry, but that’s not quite right.

Perhaps a little review will help.Perhaps a little review will help.










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Measures of Central Tendency. What Are Measures of Central Tendency?