Measures ofCentral Tendency

MeanMedianMode

What is Central Tendency?

Measures of Central Tendency:- one type of summary statistic- are measures of location within a distribu tion- A measure that tells us the middle of a bunch of data

lies - 3 most common measures of central tendency are

mean, median, mode

Importance of the Measures of Central Tendency

• To find representative value• To condense data• To make comparisons• Helpful in further statistical analysis

MEAN

The sum of a list of numbers, divided by the total number of numbers in the list.

It is the sum of all of the data values divided by the number of data values.

That is: in symbol:

Example:The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below: 15 13 18 16 14 17 12Find the mean of this set of data values.

Solution:

15

15+13+18+16+14+17+127

1057

MEDIAN

• The median (mdn) of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.

Note:• If the list has an odd number of entries, the median is

the middle entry in the list after sorting the list into increasing order.

Example: The marks of nine students in a geography test that had a maximum possible mark of 50 are given below:

47 35 37 32 38 39 36 34 35

Find the median (mdn) of this set of data values.

Solution:

Arrange the data values in order from the lowest value to the highest value:

32 34 35 35 36 37 38 39 47

The fifth data value, 36, is the middle value in this arrangement..: mdn = 36

Formula:

Note: In 32 34 35 35 36 37 38 39 47

Mdn =

Mdn =

Mdn =

12(9+1 )th𝑣𝑎𝑙𝑢𝑒

5th value

36

Note:If the list has an even number of entries, the median is

equal to the sum of the two middle (after sorting) numbers divided by two.

Example: The marks of eight students in a math test that had a maximum possible mark of 20 are given below:

12 18 16 21 10 13 17 19

Find the median of this set of data values.

Solution: Arrange the data values in order from the

lowest value to the highest value:

10 12 13 16 17 18 19 21

The number of values in the data set is 8, which is even. So, the median is the average of the two middle values.

mdn= h𝑡 𝑣𝑎𝑙𝑢𝑒

mdn=

mdm=4.5

The fourth and fifth scores, 16 and 17, are in the middle. That is, there is no one middle value.

MODE

• For lists, the mode is the most common (frequent) value.

• A data set has no mode when all the numbers appear in the data with the same frequency.

• A data set has multiple modes when two or more values appear with the same frequency.

Example:

Find the mode of the following data set: 48 44 48 45 42 49 48

The mode is 48 since it occurs most often.

Example:

The test scores of 9 seventh grade students are listed below. Find the mode.

82, 92, 75 , 91, 92, 89, 95, 100, 86

The mode is 92 since it occurs most often.

Note:

• It is possible for a set of data values to have more than one mode.

• If there are two data values that occur most frequently, we say that the set of data values is bimodal.

• If there is no data value or data values that occur most frequently, we say that the set of data values has no mode.