DATA ANALYSIS: Measures of Central Tendency & Range

DATA ANALYSIS

Measures of Central Tendency & Range

Measures of Central Tendency

​What is it?

A measure that represents the approximate centre of a dataset. Examples include: mean, median, and mode.

Mean

​a.k.a, average

The mean represents the average value of the dataset. It is calculated as follows:

​Used for QUANTITATIVE data

STEP①:	Find the sum of all the data values in the dataset.
​STEP②:	​Divide the sum by the number of data values in the data set. The quotient is the mean.

The mean of this dataset is 30. Notice, the dataset did not have to be arranged in order and the mean did not have to be one of the data values.

​Summary:

STEP①: 20 + 10 + 60 = 90
STEP②: 90 ÷ 3 = 30

​What is the mean of the following dataset:
{20, 10, 60}?

Example:

Multiple Choice

TRUE or FALSE? When calculating the mean of a dataset, you must arrange the data values in order.

• True

• False

Multiple Choice

TRUE or FALSE? It is possible to find the mean of qualitative data.

• True

• False

Multiple Choice

TRUE or FALSE? The mean does not have to be one of the data values in the dataset.

• True

• False

Multiple Choice

The height of three basketball players is 210 cm, 220 cm, and 191 cm. What is their average height?

• 200 cm

• 207 cm

• 210 cm

• 220 cm

Multiple Choice

Mr. H. gives his class a math test. The results are: 34%, 44%, 75%, 21%, 98%, 86%, 71%, 76%, 63%, and 55%.

How many students scored above the mean percentage?

• 4

• 5

• 6

• 7

Multiple Choice

To find the mean of a dataset of numbers, add up all the numbers and divide by...

• the number of data values.

• the greatest data value.

• the smallest data value.

• the middle data value.

Median

​a.k.a, middle

The median is the middle data value in an ordered dataset. It is calculated as follows:

​Used for QUANTITATIVE data

STEP①:	Arrange the data values in ascending or descending order.
​STEP②:	If the dataset has an odd number of data values, the median is simply the middle data value. However, if the dataset has an even number of data values, the median is the mean of the two middle data values.

The median of this dataset is 56. Because there is an odd number of data values, there is only one number in the middle when the dataset is ordered.

​Summary:

STEP①: 26, 39, 42, 56, 64, 71, 85

STEP②: 26, 39, 42, 56, 64, 71, 85

​What is the median of the following dataset:
{85, 42, 64, 71, 39, 56, 26}?

Example①:

​Odd Number of Data Values

​◯

The median of this dataset is 30. Because there is an even number of data values, there are two numbers in the middle when the dataset is ordered (23 and 37). Therefore, the median is the mean of these two middle numbers.

​Summary:

STEP①: 11, 13, 23, 37, 45, 58

STEP②: 11, 13, 23, 37, 45, 58

STEP③: (23 + 37) ÷ 2 = 60 ÷ 2 = 30

​What is the median of the following dataset:
{23, 37, 11, 58, 13, 45}?

Example②:

​Even Number of Data Values

​▭

Multiple Choice

When finding the median of a dataset, what do you do if there are two middle numbers?

• Write them both as the answer.

• Cry while cuddling your teddy bear.

• Pick one.

• Add them together and divide by 2.

Multiple Choice

TRUE or FALSE? Before finding the median, the dataset should be ordered first.

• True

• False

Multiple Choice

TRUE or FALSE? It is possible for the median to NOT be one of the values in the dataset.

• True

• False

Multiple Choice

The science scores of nine students in a class are 72, 94, 56, 71, 88, 95, 59, 98 and 60.

What is the median of the scores?

• 56

• 72

• 77

• 75.5

Mode

​a.k.a, most of

The mode is the data value (or data values) that occurs the most in the dataset. It is determined as follows:

​Used for QUALITATIVE or QUANTITATIVE data

STEP①:	Arrange the data values in ascending or descending order. This step is not necessary, but makes it easier when there are many data values in the dataset.
​STEP②:	Then, identify the data value (or data values) that occur the most often.

Important Considerations

It is possible for a dataset to have no mode, one mode, or modes.

​Mode Possibilities

NO MODE:	If none of the data values occur more than another, then the dataset has no modes.
​ONE MODE:	​If only one of the data values occurs more than any of the other data values in the dataset, then that data value is the only mode.

Important Considerations

​Mode Possibilities

MODES:	If more than one data value occurs the same amount of times, but more than at least one other data value in the dataset, then each of those data values is a mode.
IMPORTANT NOTE:	​Unlike the mean or median, the mode (or modes) has to be one of the data values. Also, the mean or median can only be ONE value.

No data value occurs more than another in the dataset.

​Reason:

​The following datasets have no modes.

{3, 5, 10, 6, 7, 8}

{L, M, S, XS, XL, XXL}

{2, 3, A, A, 3, 4, 4, 2}

Example①:

​No Mode

In the first dataset, 5 occurs twice, which is more than the other data values. The mode is 5.

In the second dataset, M occurs twice, but XS occurs three time. Therefore, XS is the only mode.

​Reason:

​The following datasets have one mode.

{3, 5, 5, 10, 6, 7, 8}

{L, M, M, S, XS, XS, XS, XL, XXL}

Example②:

​One Mode

In the first dataset, 5 and 10 occur twice, which is more than the other data values.
In the second dataset, L, M, and S occur three times, which is more than XS and XL. Therefore, L, M, and S are the modes.

​Reason:

​The following datasets have more than one mode.

{3, 5, 5, 10, 10, 6, 7, 8}

{L, L, L, M, M, M, S, S, S, XS, XL}

Example③:

​Modes

Multiple Choice

TRUE or FALSE? It is possible for a dataset to have no mode.

• True

• False

Multiple Choice

TRUE or FALSE? If two data values occur the most in a dataset, the mode is the mean of these two numbers.

• True

• False

Multiple Choice

Which of the following best represents the mode?

• Average

• Middle

• Most of

• I like playing games in easy mode, because it makes me feel good about myself.

Multiple Select

What is the mode of the following dataset?

{7, 14, 20, 3, 7, 14, 2, 14, 7}

• No mode

• 7

• 14

• 3

• 20

Multiple Select

What is the mode of the following dataset?

{93, 100, 82, 93, 82, 100, 100, 93, 82, 71, 71, 71}

• No mode

• 82

• 93

• 100

• 71

Multiple Select

What is the mode of the following dataset?

{😎💚👽👽👽😎😎🌼💚🌼💚}

• No mode

• 😎

• 💚

• 👽

• 🌼

Range

​a.k.a, spread of the data

The range of a dataset is the difference between the largest and smallest data values. It gives us an idea of how much variation there is within the data. It is determined as follows:

​Used for QUANTITATIVE data

STEP①:	Largest data value - Smallest data value
​STEP②:	The difference is the range.

​What is the range of the following dataset?
{3, 5, 10, 6, 7, 8}







∴ the range of this dataset is 7.

Example:

​Calculating the Range

RANGE =	Largest data value - Smallest data value
​RANGE = RANGE =	10 - 3 7

Multiple Choice

TRUE or FALSE? The range of a dataset is a measure of central tendency.

• True

• False

Multiple Choice

TRUE or FALSE? To find the range of a dataset, you must arrange the data values in order.

• True

• False

Multiple Choice

TRUE or FALSE? The range does not have to be one of the data values in the dataset.

• True

• False

Multiple Choice

Which of the following is the correct way to calculate the range of a dataset?

• mean subtract median

• smallest data value subtract largest data value

• (base x height) ÷ 2

• find the mean of the largest and smallest data values

• none of these

Additional Resources:

• Math Antics → https://www.youtube.com/watch?v=B1HEzNTGeZ4

• Math with Mr. J. → https://www.youtube.com/watch?v=e3uY2LraXts

• GoNoodle → https://www.youtube.com/watch?v=LeKbUZ954lE