Measures of Central Tendency

Mean,Median and Modefor Ungrouped Data

Basic Statistics

Measures of Central Tendency

In layman’s term, a measure of central tendency is an AVERAGE. It is a single number of value which can be considered typical in a set of data as a whole.

For example, in a class of 40 students, the average height would be the typical height of the members of this class as a whole.

Measures of central tendency are numerical

descriptive measures which indicate or

locate the center of a distribution or

data set.

MEAN

Among the three measures of central tendency, the mean is the most popular and widely used. It is sometimes called the arithmetic mean.

If we compute the mean of the population, we call it the parametric or population mean, denoted by μ (read “mu”).

If we get the mean of the sample, we call it the sample mean and it is denoted by (read “x bar”).

The MEAN of a set of values or measurements is the sum of all the measurements divided by the number of measurements in the set.

Mean for Ungrouped Data

Example 1:Ms. Sulit collects the data on the ages of Mathematics teachers in Santa Rosa School, and her study yields the following:

38 35 28 36 35 3340Solution:

= 35

Based on the computed mean, 38 is the average age of Mathematics teachers in SRS.

For ungrouped or raw data, the mean has the following formula.

where = mean = sum of the measurements or values n = number of measurements

Your turn!

Mang John is a meat vendor. The following are his sales for the past six days. Compute his daily mean sales.

Tuesday P 5 800Wednesday 8 600Thursday 6 500Friday 4 300

Saturday 12 500Sunday 13 400

Solution:

= 51, 100

The average daily sales of Mang John is P51,100.

Weighted Mean

Weighted mean is the mean of a set of values wherein

each value or measurement has a different weight or

degree of importance. The following is its formula:

where = mean

x = measurement or value

w = number of measurements

ExampleBelow are Amaya’s subjects and the corresponding number

of units and grades she got for the previous grading period. Compute her grade point average.

Subject Units Grade

Filipino .9 86

English 1.5 85

Mathematics 1.5 88

Science 1.8 87

Social Studies .9 86

TLE 1.2 83

MAPEH 1.2 87

= 86.1

Amaya’s average grade is 86.1

Your turn!James obtained the following grades in his five subjects for the second grading period. Compute his grade point average.

Solution:

= 87.67

James general average is 87.67

Subject Units Grade

Math 1.5 90

English 1.5 86

Science 1.8 88

Filipino 0.9 87

MAKABAYAN 1.5 87

Likert-type Question

Next are examples of Likert-type statements. Respondents will choose the number which best represents their feeling regarding the statements. Note that the statements are grouped according to a theme.

This is used if the researcher wants to know the feelings or opinions of the respondents regarding any topic or issues of interest.

Choices

5 (SA) Strongly Agree

4 (A) Agree

3 (N) Neutral

2 (D) Disagree

1 (SD) Strongly Disagree

Students’ personal confidence in learning Statistics

5 4 3 2 1

1. I am sure that I can learn Statistics

2. I think I can handle difficult lessons in Statistics.

3. I can get good grades in Statistics.

Source: B.E. Blay, Elementary Statistics

Below are the responses in the Likert-type of statements above. The table below shows the mean responses and their interpretation. Using the formula for computing the weighted mean, check the correctness of the given means on the table.

5 4 3 2 1 Mean Interpretation

1 36 51 18 0 1 4.14 Agree

2 18 44 37 8 1 3.65 Agree

3 18 48 28 0 1 3.86 Agree

Likert-type Mean Interpretation

1.0 - 1.79 - Strongly Disagree

1.8 - 2.59 - Disagree

2.6 - 3.39 - Neutral

3.4 - 4.19 - Agree

4.2 - 5.00 - Strongly Agree

Your turn!Below is the result of the responses to the following Likert-type statements . Solve for the mean and give the interpretation.

5 4 3 2 1 Mean Interpretation

1 33 49 26 1 1

2 35 45 31 0 1

3 34 58 21 0 0

Students’ perception on Statistics as a subject

5 4 3 2 1

1. I think Statistics is a worthwhile, necessary subject

2. I will use Statistics in many ways as a professional

3. I’ll need a good understanding of Statistics for my research work

Properties of Mean

1. Mean can be calculated for any set of numerical data, so it always exists.

2. A set of numerical data has one and only one mean.

3. Mean is the most reliable measure of central tendency since it takes into account every item in the set of data.

4. It is greatly affected by extreme or deviant values (outliers)

5. It is used only if the data are interval or ratio.

MEDIAN

The MEDIAN, denoted Md, is the middle

value of the sample when the data are ranked in

order according to size.

16 17 18 19 20 21 22

Median

16 17 18 19 20 21 22 23

Median

Your turn!

Compute the median and interpret the result.1. In a survey of small businesses in Tondo, 10 bakeries

report the following numbers of employees: 15, 14, 12, 19, 13, 14 15, 18, 13, 19.

2. The random savings of 2nd year high school students reveal the following current balances in their bank accounts:

3. The following are the lifetimes of 9 lightbulbs in thousands of hours.

Students A B C D E F G H

Current Balances P340 350 450 500 360 760 800 740

Lightbulb A B C D E F G H I

Lifetime 1.1 1.1 1.2 1.1 1.4 .9 .2 1.2 1.7

Properties of Median

1. Median is the score or class in the distribution wherein 50% of the score fall below it and another 50% lie.

2. Median is not affected by extreme or deviant values.

3. Median is appropriate to use when there are extreme or deviant values.

4. Median is used when the data are ordinal.

5. Median exists in both quantitative or qualitative data.

MODE

The MODE, denoted Mo, is the value which

occurs most frequently in a set of measurements or

values. In other words, it is the most popular value

in a given set.Examples:Find the Mode.1. The ages of five students are: 17, 18, 23, 20, and 19 2. The following are the descriptive evaluations of 5

teachers: VS, S, VS, VS, O3. The grades of five students are : 4.0, 3.5, 4.0, 3.5, and

1.04. The weights of five boys in pounds are: 117, 218, 233,

120, and 117

Properties

1. It is used when you want to find the value which occurs most often.

2. It is a quick approximation of the average.

3. It is an inspection average.

4. It is the most unreliable among the three measures of central tendency because its value is undefined in some observations.

Your turn!

Find the mode and interpret it.

1. The following table shows the frequency of errors committed by 10 typists per minute.

2. A random sample of 8 mango trees reveals the following number of fruits they yield

3. The following are the scores of 9 students in a Mathematics quiz.: 12, 15, 12, 8, 7, 15, 19, 24, 13

Mango Tree A B C D E F G H

No. of fruits 80 70 80 90 82 82 90 82

Typists A B C D E F G H I J

No. of errors per min. 5 3 3 7 2 8 8 4 7 10