CHAPTER 3
DATA DESCRIPTION
LESSON 1 -Measures of Central Tendency
INTRODUCTION:
Statistical methods are needed for summarizing and describing gathered
numerical data. However, by looking at the tables and graphs, one can have a
difficulty in describing the entire set of data. Better, if we can pick or choose a
single score that would represent the entirety of the data set. This value will be
helpful in making decisions based on the data collected. The measures of
central tendency are used to select the central value that could represent the
entire set of data. Thus, helping the investigator in decision making.
OBJECTIVES:
At the end of this lesson, you should be able to:
1. Describe the uses of the measures of central tendency;
2. Compute and interpret the mean, median and mode; and
3. Discuss the properties of mean, median and mode.
The Mean
The mean is the arithmetic average of the all the scores in the data set.
This is the most frequently used measure of central tendency. This is used when
the data is either interval or ratio.
The symbol x read as “x bar” is used to represent sample mean while μ
read as “mu” represents the population mean.
Properties of Mean
➢ The most reliable measure of central tendency.
➢ Used when the data is interval or ratio.
➢ It is the layman’s concept of the average.
➢ All observations are considered in its computation.
➢ It is markedly affected by extreme values
➢ Used when the distribution is normal or is not badly skewed.
The Mean of ungrouped data
The mean of ungrouped data can be determined by adding the all the
scores or data and divide the sum by the numbers of scores in the data. In
symbol,
n
x + x 2 + x 3 + ... + x n x i
x= 1 or i=1
n n
, Solution:
5 + 7 + 9 + 10 + 12 + 15 58
x= = = 9.67
6 6
The Mean Using MS Excel
Step 1. Open your MS Excel program.
Step 2. Type in your data.
Step 3. Select the box where you want to put the mean.
Step 4. Type the formula.
=AVERAGE(range of your data)
Step 5. Press “Enter”
You will get mean equals 9.666667 which yields the same result.
Example 2. Aling Nilda is a “kuron” vendor. In a week she earned in her
business the following amount 450, 525, 300, 740, 675, 800 and 270. Find the
mean sale of Aling Nilda in a week.
Solution:
450 + 525 + 300 + 740 + 675 + 800 + 270 3760
x= = = 537.14
7 7
This means that Aling Nilda has an average earning of Php 537.14 in a week,
The Weighted Mean
There are times that a number has a certain weight. For example, you
are asked to determine your mean grade in the first semester. Given the fact
that every course (subject) has a weight, this can be done by getting the sum of
the products of a number and its weight divided by the total weight.
Suppose, x1, x2, x3 , … , xn are the scores and their respective weights are
w1, w2, w3, … , wn, the weighted mean of the scores is defined as
x 1 w 1 + x 2 w 2 + x 3 w 3 + ... + x n w n
x=
w 1 + w 2 + w 3 + ... + w n
Example 3. John’s grades last semester are as follow:
DATA DESCRIPTION
LESSON 1 -Measures of Central Tendency
INTRODUCTION:
Statistical methods are needed for summarizing and describing gathered
numerical data. However, by looking at the tables and graphs, one can have a
difficulty in describing the entire set of data. Better, if we can pick or choose a
single score that would represent the entirety of the data set. This value will be
helpful in making decisions based on the data collected. The measures of
central tendency are used to select the central value that could represent the
entire set of data. Thus, helping the investigator in decision making.
OBJECTIVES:
At the end of this lesson, you should be able to:
1. Describe the uses of the measures of central tendency;
2. Compute and interpret the mean, median and mode; and
3. Discuss the properties of mean, median and mode.
The Mean
The mean is the arithmetic average of the all the scores in the data set.
This is the most frequently used measure of central tendency. This is used when
the data is either interval or ratio.
The symbol x read as “x bar” is used to represent sample mean while μ
read as “mu” represents the population mean.
Properties of Mean
➢ The most reliable measure of central tendency.
➢ Used when the data is interval or ratio.
➢ It is the layman’s concept of the average.
➢ All observations are considered in its computation.
➢ It is markedly affected by extreme values
➢ Used when the distribution is normal or is not badly skewed.
The Mean of ungrouped data
The mean of ungrouped data can be determined by adding the all the
scores or data and divide the sum by the numbers of scores in the data. In
symbol,
n
x + x 2 + x 3 + ... + x n x i
x= 1 or i=1
n n
, Solution:
5 + 7 + 9 + 10 + 12 + 15 58
x= = = 9.67
6 6
The Mean Using MS Excel
Step 1. Open your MS Excel program.
Step 2. Type in your data.
Step 3. Select the box where you want to put the mean.
Step 4. Type the formula.
=AVERAGE(range of your data)
Step 5. Press “Enter”
You will get mean equals 9.666667 which yields the same result.
Example 2. Aling Nilda is a “kuron” vendor. In a week she earned in her
business the following amount 450, 525, 300, 740, 675, 800 and 270. Find the
mean sale of Aling Nilda in a week.
Solution:
450 + 525 + 300 + 740 + 675 + 800 + 270 3760
x= = = 537.14
7 7
This means that Aling Nilda has an average earning of Php 537.14 in a week,
The Weighted Mean
There are times that a number has a certain weight. For example, you
are asked to determine your mean grade in the first semester. Given the fact
that every course (subject) has a weight, this can be done by getting the sum of
the products of a number and its weight divided by the total weight.
Suppose, x1, x2, x3 , … , xn are the scores and their respective weights are
w1, w2, w3, … , wn, the weighted mean of the scores is defined as
x 1 w 1 + x 2 w 2 + x 3 w 3 + ... + x n w n
x=
w 1 + w 2 + w 3 + ... + w n
Example 3. John’s grades last semester are as follow:
