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SECTION IV
Measures of Central Tendency and Measures of Location
Target Outcomes
At the end of the lesson, you are expected to:
- Compute the mean, median, and mode of grouped and ungrouped data;
- interpret the measures of central tendency; and
- determine the deciles, percentiles, and quartiles of given grouped and ungrouped data using formulas.
Abstraction
The Frequency Distribution Table
The frequency distribution table shows the number of items falling into each group. In an experiment, various items of a series are classified into groups. As an example, suppose you recorded the weight (in kilograms) of swine after 90 days of feeding them formulated feeds.
The raw data is the set of data in its original form. An array is an arrangement of observations according to their magnitude, either increasing or decreasing. The following tables are the raw data and array gathered from the experiment. Let us name this data set, data_swine.
82 82 83 79 72 71 84 59 77 50 87 50 53 63 69 72 74 76 79 82 84 87 83 82 63 75 50 85 76 79 68 69 62 50 57 65 69 72 74 77 80 82 84 87 79 69 74 53 73 71 50 76 57 81 62 50 59 66 69 72 75 77 80 82 84 88 71 88 84 80 68 50 74 84 71 73 68 50 59 66 69 72 75 77 80 82 84 89 71 80 72 50 76 89 94 80 84 81 50 50 60 68 70 72 75 77 81 83 85 89 84 76 75 82 72 53 91 69 60 89 79 50 60 68 71 73 75 78 81 83 85 91 59 62 79 82 81 81 60 84 68 66 94 50 62 68 71 73 75 79 81 84 86 92 77 78 87 75 86 82 74 73 72 84 51 51 62 68 71 73 76 79 81 84 86 94 50 69 75 70 77 72 86 77 75 96 66 52 62 68 71 73 76 79 82 84 87 94 87 73 84 68 85 62 87 92 69 52 65 53 62 69 71 74 76 79 82 84 87 96
The following are definition of terms that we will use in the construction of the frequency distribution table.
- Class Frequency – the number of observation falling in the class
- Class Interval – the numbers defining the class
- Class limits – the end numbers of the class
- Class boundaries – the true class limits.
LCB (lower class boundary) – is usually defined as halfway between the lower class limit of the class and the upper class limit of the preceding class.
UCB (upper class boundary) – is usually defined as halfway between the upper class limit of the class and the lower class limit of the next class.
- Class size – is the number of values in the class interval. It can be computed as the difference between the upper class boundaries of the class and the preceding class; can also be computed as the difference between the lower class boundaries of the current class and the next class; can also be computed by using the respective class limits instead of the class boundaries.
- Class mark (CM) – midpoint of a class interval.
- Open-end class – a class that has no lower limit or upper limit.
Steps in Constructing a Frequency Distribution Table
Step 1: Determine the number of classes, k. There must be an adequate number of classes to show the essential characteristics of the data; at the same time, there should not be too many classes that it is already difficult to grasp the picture of the distribution as a whole. There are no precise rules concerning the optimal number of classes.
Step 2: Determine the approximate class size. Whenever possible, all classes should be of the same size. The following are steps to follow in determining the class size. Solve for the range, R = max. – min. Compute for C’ = R ÷ k Round- off C’ to a convenient number to work with, say c , and use c as the class size.
Step 3: Determine the lowest class limit. The first class must include the smallest value in the data set. The class size determines the class interval.
Step 4: Determine all class limits by adding the class size, c , to the limit of the previous class.
Measures of Central Tendency
A measure of central tendency is any single value that is used to identify the “center” or the typical value of a data set. The following are characteristics of the measures of central tendency.
Characteristics of the Mean
- It is the most familiar measure used, and it employs all available information.
- It is affected by the value of every observation. In particular, it is strongly influenced by extreme values.
- Since the mean is a calculated number, it may not be an actual number in the data set.
- It possesses two mathematical properties that will prove to be important in subsequent analyses. The sum of the deviations of the values from the mean is zero. The sum of the squared deviations is minimum when the deviations are taken from the mean.
- As a remark, (a) if a constant c is added (or subtracted) to all observations, the mean of the new observations will increase (decrease) by the same amount c; (b) if all observations are multiplied or divided by a constant, the new observations will have a mean that is the same constant multiple of the original mean.
Characteristics of the Median
- The median is a positional measure
- The median is affected by the position of each item in the series but not by the value of each item. This means that extreme values affect the median less than the arithmetic mean.
Characteristics of the Mode
- It does not always exist; and if it does, it may not be unique. A data set is said to be unimodal if there is only one mode, bimodal if there are two modes, trimodal if there are three modes, and so on.
- It is not affected by extreme values.
- The mode can be used for qualitative as well as quantitative data.
The lessons in this section discuss the formula to be used in dealing with grouped or ungrouped data. We will also provide examples on how we will use the formula to arrive at appropriate measures to describe the results of our study. Lessons 1 and 2 deal with the measures of central tendency given an ungrouped and grouped data. Lesson 3 just recognizes the mean used in grouped and ungrouped data. Lastly, Lesson 4 talks about the measures of location of a given data set.
LESSON 1. MEAN, MEDIAN AND MODE OF RAW (UNGROUPED) DATA
A. Mean of a raw (ungrouped) data
The mean of a raw (ungrouped data) is given by the sum of all the values of the observations divided by the number of observations. In symbols, we have
where:
𝑋̅ – is the mean of an ungrouped data
𝑋𝑖 – are the observations in a given experiment
n – is the number of observations in the experiment
Example 1:
Consider again our data set data_swine. Using (1) above, we have
Example 2:
The numbers of employees at 5 different food companies are 10, 12, 6, 8, and 4. Find the mean number of employees for the 5 stores.
Example 3:
Percentage scores in Applied Statistics Midterm Exam for a sample of 10 students are as follows: 60, 55, 30, 90, 88, 79, 45, 66, 93, and 80. Find the mean.
82 + 82 + 83 + 79 + 72 + 71 + 84 + 59 + 77 + 50 + 87 + 83 + 82 + 63 + 75 + 50 + 85 + 76 + 79 + 68 + 69 + 62 + 79 + 69 + 74 + 53 + 73 + 71 + 50 + 76 + 57 + 81 + 62 + 71 + 88 + 84 + 80 + 68 + 50 + 74 + 84 + 71 + 73 + 68 + 71 + 80 + 72 + 50 + 76 + 89 + 94 + 80 + 84 + 81 + 50 + 84 + 76 + 75 + 82 + 72 + 53 + 91 + 69 + 60 + 89 + 79 + 59 + 62 + 79 + 82 + 81 + 81 + 60 + 84 + 68 + 66 + 94 + 77 + 78 + 87 + 75 + 86 + 82 + 74 + 73 + 72 + 84 + 51 + 50 + 69 + 75 + 70 + 77 + 72 + 86 + 77 + 75 + 96 + 66 + 87 + 73 + 84 + 68 + 85 + 62 + 87 + 92 + 69 + 52 + 65
Example 3:
50 53 63 69 72 74 76 79 82 84 87 50 57 65 69 72 74 77 80 82 84 87 50 59 66 69 72 75 77 80 82 84 88 50 59 66 69 72 75 77 80 82 84 89 50 60 68 70 72 75 77 81 83 85 89 50 60 68 71 73 75 78 81 83 85 91 50 62 68 71 73 75 79 81 84 86 92 51 62 68 71 73 76 79 81 84 86 94 52 62 68 71 73 76 79 82 84 87 94 53 62 69 71 74 76 79 82 84 87 96
𝟐
𝟐+𝟏 𝟐
𝟐
𝟐+𝟏 𝟐
C. Mode of a raw (ungrouped) data
The observed value that occurs most frequently is called the mode of the data set. In other words, it is a value that occurs with the greatest density. Generally, it is a less popular measure than the mean and the median. It is determined by counting the frequency of each value and finding the value with the highest frequency of occurrence.
Examples:
- 2, 5, 2, 3, 5, 2, 1, 4, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2
1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5 Mode = 2
- 2, 5, 5, 2, 2, 5, 1, 3, 5, 4, 2, 5, 5, 2, 2, 5, 5, 2, 2, 1
1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5 Mode = 2, 5 (bimodal because both 2 and 5 have the highest frequency)
- 1, 2, 3, 3, 2, 1, 2, 3, 1, 4, 4, 5, 5, 1, 2, 3, 4, 5, 4, 5
1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5 No mode. 1, 2, 3 4, 5 all have the same frequency. No number is any more common than any other.
LESSON 2. MEAN, MEDIAN AND MODE OF GROUPED DATA
In this lesson, we will approximate the mean, median, and mode of a group data from the frequency distribution. In our examples below, we will consider again our data set, data_swine and its frequency distribution table in Table 1.
A. Mean of a Grouped Data
In approximating the mean from a frequency distribution, we have to take note that when the class mark is assumed to be a representative of all the values in the class, the following equation may be used to approximate the mean of a grouped data from a frequency distribution.
where:
𝑿̅ – is the mean of a grouped data
𝒇𝒊 – is the frequency of the ith class
𝑿𝒊 – is the class mark of the ith class
n = ∑ 𝒏𝒊=𝟏 𝒇𝒊– is the total number of observations
Example:
Class f (𝒇𝒊)
CM
50 - 54 11 52 572 55 - 59 3 57 171 60 - 64 7 62 434 65 - 69 13 67 871 70 - 74 18 72 1 296 75 - 79 19 77 1 463 80 - 84 23 82 1 886 85 - 89 11 87 957 90 - 94 4 92 368 95 - 99 1 97 97 Total 110 8145
C. Mode of a Grouped Data
In approximating the mode from a frequency distribution, the following procedure may be followed to approximate the mode of a grouped data from a frequency distribution.
Step 1: In the frequency distribution table, locate the modal class. The modal class is the class with the highest frequency.
Step 2: Approximate the mode using the following formula
where:
𝑳𝑪𝑩𝒎𝒐 – is the lower class boundary of the modal class
C – is the class size of the modal class
𝒇𝒎𝒐 – is the frequency of the modal class
𝒇𝟏 – is the frequency of the class preceding to the modal class
𝒇𝟐 – is the frequency of the class following the modal class
Example:
**Class f LCB UCB CM RF RFP >CF
LESSON 3. WEIGHTED MEAN, ARITHMETIC MEAN
A. Arithmetic Mean
The arithmetic mean is just the mean described in the mean of an ungrouped data. It is generally given by equation (1).
The population mean for a finite population with N elements, denoted by the Greek letter 𝜇 (mu), is computed as
The sample mean , 𝑿̅ (read as “X bar”) of n observations is computed as
The sample mean (a statistic) is an estimate of the unknown population mean (a parameter).
B. Weighted Mean
The weighted mean is a modification of the usual mean that assigns weights (or measure of relative importance) to the observations to be averaged. It is just the mean described in Lesson 2A. If each observation, 𝑿𝒊, is assigned a weight 𝑾𝒊, 𝒊 = 𝟏, 𝟐, 𝟑, … , 𝒏, the weighted mean is given by
27.75𝑡ℎ^ = 27𝑡ℎ^ + 0.75(28𝑡ℎ^ − 27𝑡ℎ)
In the data set, 27.75𝑡ℎ^ means the value is located between the 27 𝑡ℎ^ and the 28 𝑡ℎ value. Hence, the 27 𝑡ℎ^ value is 68. Now, 0.75 must be multiplied to the difference of the value in the 27 𝑡ℎ^ and 28 𝑡ℎ^ place, which in our array, is both 68.
For 𝑸𝟐 ,
𝑡ℎ
𝑡ℎ = 55.5𝑡ℎ
is the position of the second quartile in the array. That is, the value of the second quartile is
55.5𝑡ℎ^ = 55𝑡ℎ^ + 0.5(56𝑡ℎ^ − 55𝑡ℎ)
= 75 + 0.5 (75 − 75)
= 75.
In the data set, 55.5𝑡ℎ^ means the value is located between the 55 𝑡ℎ^ and the 56 𝑡ℎ value. Hence, the 55 𝑡ℎ^ value is 75. Now, 0.5 must be multiplied to the difference of the value in the 55 𝑡ℎ^ and 56 𝑡ℎ^ place, which in our array, is both 75.
For 𝑸𝟑 ,
𝑡ℎ
𝑡ℎ = 83.25𝑡ℎ
is the position of the third quartile in the array. That is, the value of the third quartile is
83.25𝑡ℎ^ = 83𝑟𝑑^ + 0.25(84𝑡ℎ^ − 83𝑟𝑑)
= 82 + 0.25 (82 − 82)
= 82.
In the data set, 83.25𝑡ℎ^ means the value is located between the 83 𝑟𝑑^ and the 84 𝑡ℎ^ value. Hence, the 83 𝑟𝑑^ value is 82. Now, 0.25 must be multiplied to the difference of the value in the 83 𝑟𝑑^ and 84 𝑡ℎ^ place, which in our array, is both 82.
To compute for the ith quartile in a grouped data , we have
𝒊𝒏 𝟒 − <𝑪𝑭𝑳𝑪𝑩𝑸𝒊−𝟏 𝒇𝑸𝒊
where:
𝒇𝑸𝒊 – is the frequency of the quartile class
< 𝑪𝑭𝑳𝑪𝑩𝑸 𝒊−𝟏^
- is the CF To compute for the ith quartile in a grouped data , we have
𝒊𝒏 𝟏𝟎 − <𝑪𝑭𝑳𝑪𝑩𝑫𝒊−𝟏 𝒇𝑫𝒊
where:
𝒇𝑫𝒊 – is the frequency of the decile class
< 𝑪𝑭𝑳𝑪𝑩𝑫𝒊−𝟏 – is the CF Step 2: Use formula (13) to compute for 𝐷𝑖
𝑖𝑛 10 − <𝐶𝐹𝐿𝐶𝐵𝐷 8 − 𝑓𝐷8^ ) 𝑐 = 79.5 + (
8(110) 10 − 71 23 ) 10
Exercise:
Find 𝐷 1 , 𝐷 2 , 𝐷 3 , 𝐷 4 , 𝐷 5 , 𝐷 6 , 𝐷 7 , 𝐷 9.
C. PERCENTILES
Percentiles are values that divide a set of observations in an array into 100 equal parts. Thus,
𝑷𝟏 is read as first percentile. It is the value below which 1% of the values fall.
𝑷𝟐 is read as second percentile. It is the value below which 2% of the values fall.
𝑷𝟐 is read as second percentile. It is the value below which 2% of the values fall.
…
𝑷𝟗𝟗 is read as ninety-ninth percentile. It is the value below which 99% of the values fall.
To compute for the position of the ith decile in an ungrouped data , we have
Example:
Using the data_swine , find 𝑷𝟓𝟒.
𝑡ℎ
𝑡ℎ = 59.94𝑡ℎ
is the position of the 54 𝑡ℎ^ percentile in the array. That is, the value of the 54 𝑡ℎ^ percentile is
59.94𝑡ℎ^ = 59𝑡ℎ^ + 0.94(60𝑡ℎ^ − 59𝑡ℎ)
= 76 + 0.94 (76 − 76)
= 76.
Step 1: Identify the percentile class.
𝒊𝒏 𝟏𝟎𝟎 =^
𝟑𝟒(𝟏𝟏𝟎) 𝟏𝟎𝟎 = 𝟑𝟕. 𝟒 The
