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cies Concept of Central
UNIT 1 CONCEPT OF CENTRAL TENDENCY^ Tendency
Structure 1.0 Introduction 1.1 Objectives 1.2 Meaning of Measures of Central Tendency 1.3 Functions of Measures of Central Tendency 1.4 Characteristics of a Good Measures of Central Tendency 1.5 Types of Measures of Central Tendency 1.5.1 The Mean 1.5.1.1 Properties of the Mean 1.5.1.2 Limitations of the Mean 1.5.2 The Median 1.5.2.1 Properties of the Median 1.5.2.2 Limitations of the Median 1.5.3 The Mode 1.5.3.1 Properties of the Mode 1.5.3.2 Limitations of the Mode 1.6 Let Us Sum Up 1.7 Unit End Questions 1.8 Glossary 1.9 Suggested Readings
1.0 INTRODUCTION
In the day to day situations you must have heard that the average height of the Indian boys is 5 feet10 inches. The average longevity of Indians has now increased the average number of working women have gone up in the last ten years. Ever wondered what this ‗average‘ is all about. This is nothing but the measure of central tendency. Tests and experiments and survey studies in psychology provide us data mostly in the shape of numerical scores. In their original form these data have little meaning for the investigator or reader. The obtained data are complicated and scattered and no inference can be arrived at, unless they are organised and arranged in some systematic way.
The classifications and tabulation makes the data easy and clear to understand. In the previous units you have learned about how you can classify and organise the data as well you had learned about presentation of data in the forms of graph. But you want to describe a data. A useful way to describe a data as a whole is to find a single number that represents what is average or typical of that set of data. This can be obtained by way of measures of central tendency, because it is generally located toward the middle or center of a distribution, where most of the data tend to be concentrated.
In this unit we will learn about measures of central tendency. There are many measures of central tendency, we will focus on only the three most commonly used in psychology, i.e. arithmetic mean, median and mode. We will learn about their properties and their limitations.
Central Tendencies and Dispersion
Con
1.1 OBJECTIVES
After reading this unit, your will be able to understand: Describe the meaning of measures of central tendency; State the functions of measures of central tendency; Name the different types of measures of central tendency; and Explain the properties and limitations of mean, median, and mode.
1.2 MEANING OF MEASURES OF CENTRAL
TENDENCY
The term central tendency was coined because observation (numerical value) in most data show a distinct tendency of the group to cluster around a value of an observation located some where in the middle of all observations. It is necessary to identify or calculate this typical central value to describe the characteristics of the entire data set. This descriptive value is the measure of central tendency and methods of compiling this central value are called measures of central tendency.
According to English & English (1958) ―Measure of a central tendency is a statistic calculated from a set of distinct and independent observations and measurements of a certain items or entity and intend to typify those observations.
According to Chaplin (1975) ―Central tendency refers to the representative value of the distribution of scores‖.
If we take the achievement scores of the students of a class and arrange them in a frequency distribution, we may sometimes find that there are very few students who either score very high or very low. The marks of most of the students lie somewhere between the highest and the lowest scores of the whole class. This tendency of the data to converge around a distribution, named as central tendency.
1.3 FUNCTIONS OF MEASURES OF CENTRAL
TENDENCY
Measure of central tendency provides us a single summary figure that best describes the central location of an entire distribution of observation.
It is helpful in reducing the large data into a single value. For example , it is difficult to know an individual family‘s need for water during summer in cities. But the knowledge of the average quantity of water needed for the entire population of a city, will help the water works departments in planning for water resources.
In psychology we draw the representative sample from the population and information are gathered regarding different attributes. The mean of the sample provide us the idea about the mean of the population.
These measures are helpful in making decisions. For example the education department may be intended to know the average number of boys and girls
Central Tendencies and Dispersion
Definitions of Mean, Mode and Median^ Con In statistics, we use a term called statistical distribution. This actually tells us how a group of data is distributed in a population. For instance if you want to know amongst the population of India, how many are male children, how many female children, how many adult males and how many adult females, all these we can represent in the data in a statistical distribution in terms of actual numbers or in terms of percentage. Which ever way we do, as we look at the graph, we will know how the males and females are distributed across the population. In this statistical distribution, one can describe the properties of the distribution in terms of mean, median, mode, and range. Thus in statistics, a distribution is the set of all possible values for terms that represent defined events.
In statistical distributions, there are actually two types, viz., (i) the discrete random variable distribution and (ii) the continuous random variable distribution.
The discrete random variable distribution means that every term in the distribution has a precise, isolated numerical value. An example of a distribution with a discrete random variable is the set of results for a test taken by a class in school. The continuous random variable distribution has generally values within an interval or span. To give an example, scores of test taken by 5 students are 45,55,59,50,40. These are independent scores and called discrete scores. If the same scores are given as 45-47, 48-50, 51-53,54- 56, 57-59, then this is called continuous random distribution. In the latter continuous distribution, a term can acquire any value within an unbroken interval or span. Such a distribution is also called a probability density function, which is used in weather forecasts.
Mean As mentioned in the introduction mean is the average of all the scores in a discrete or continuous distribution. Method of calculation of this mean is different for discrete distribution and the continuous distribution. This average so calculated is called the Mean or mathematical average. It is easier to calculate mean from discrete data by adding up all the scores of X1 to Xn and divide by X1 + X2+… / Xn. On the other hand to calculate mean from continuous distribution a formula has to be used and generally there are more than one method to calculate the mean from continuous distribution.
Median The median is the midpoint of a series of data. If it is a discrete data and having even number of scores (5,7,9, etc.) then the middle item leaving equal number of scores below and above the middle item is considered to be the median. If the discrete data is of even number, then the middle two items have to be added and divided by two to get the median. (The calculation of the median will be taken up in another unit as here we discuss the clear concept about the measures of central tendency). From a continuous distribution also median can be calculated but as mentioned in the case of calculation of mean here too certain formula has to be applied. Median thus is the 50th^ % igtem in a series whether it is discrete or continuous.
cies Concept of Central Mode Tendency When the data is arranged in a frequency, that is, for example, all the test scores obtained by 15 students in English, certain scores like 55 marks out of 100 may appear 4 times, that is 4 students would have scored 55. ( For example, 45, 55, 43,44, 55 ,45,65,63,67, 55 ,,41,42, 55 ,67.) The remaining have scored different, marks in the test. At one glance we could see that the largest number of times a score appears is 55 that is by 4 students while all other scores appear only once). Thus the mode here is 55 which has appeared the largest number of times or we can say 55 has the largest frequency of 4 students getting that mark in English. Thus one may state that the mode of a distribution with a discrete random variable is the value of the term that occurs the most often. It should also be kept in mind that there could be 4 students getting marks of 65. Thus 55 and 65 both have a frequency of 4 students each and both can be considered as the Mode. Thus it is possible that for a distribution with a discrete random variable can have more than one mode, especially if there are not many terms. A distribution with two modes is called bimodal. A distribution with three modes is called trimodal. The mode of a distribution with a continuous random variable is also calculated with the help of a formula, which will be taken up in another unit. Self Assessment Questions
- What are the types of measures of central tendency? …………………………………………………………………………….. …………………………………………………………………………….. …………………………………………………………………………….. ……………………………………………………………………………..
- Define mean and put forward the properties of the mean. What are its limitations? …………………………………………………………………………….. …………………………………………………………………………….. …………………………………………………………………………….. ……………………………………………………………………………..
- Define median and discuss its properties and limitations. …………………………………………………………………………….. …………………………………………………………………………….. …………………………………………………………………………….. ……………………………………………………………………………..
- What is mode? How will you identify mode. Discuss its properties and limitations. …………………………………………………………………………….. …………………………………………………………………………….. …………………………………………………………………………….. ……………………………………………………………………………..
cies Concept of Central
1.5 TYPES OF MEASURES OF CENTRAL Tendency
TENDENCY
There are many measures of central tendency. We will consider only the three most commonly used in psychology: The Mean, Median and Mode.
1.5.1 The Mean
This, is the most commonly used measure of central tendency. There are different types of means, that is, Arithmetic mean, Geometric mean and Harmonic mean, but we will focus only on arithmetic mean in this unit.
The arithmetic mean is the sum of all the scores in a distribution divided by the total number of scores.
M is the symbol for the Mean. M is used in research articles in psychology and are recommended by the style guidelines of the American Psychological Association (2001).
It should be noted that the Greek letter (μ ) pronounced as mue is used to denote the mean of the population and X pronounced as ―X-bar‖ or M is used to denote the mean of a sample. Let us now look at the properties of these measures of central tendencies.
1.5.1.1 Properties of the Mean
The mean is responsive to the exact position of each score. In the next unit when we will learn how to compute the mean you will see that increasing or decreasing the value of any score changes the value of the mean.
The mean is sensitive to the presence (or absence) of extreme scores. For, example. The means of the scores 6,7,9,8 is 7. 6+7+9+8 /4 = 7. 5 But the mean of scores 6,7,9,22 is 6+7+9+22 / 4 = 11 It means that one extreme value in a series of values, can make drastic changes in the mean values.
When a measure of central tendency should reflect the total of the scores, the Mean is the best choice because the Mean is the only measure based on the total scores. For example, if a teacher wants to see the effect of training on pupils performance then the teacher can get the mean scores of each student, before and after the training and compare the mean scores obtained before and after the training. The difference will give an idea of the effect of training on the students. If the difference is great then one may be able to state that the training had a good effect on the students.
The mean will best suit this kind of problems because the teacher is interested in knowing the usefulness of the training for the students. Similarly insurance company expresses life expectancy as a Mean, because on the basis of this, companies can come to know the total income from policy holders and total pay off to survivors.
Central Tendencies and Dispersion
When we have to do the further statistical computations, the mean is the most^ Con useful statistic to use.
As you will see in the next unit, Mean is based on arithmetic and algebraic manipulations, so it fits in with other statistical formulas and procedures. When we have to do further calculations, mean is often incorporated implicitly or explicitly.
1.5.1.2 Limitations of the Mean
One of the most important limitation of the mean it is too sensitive to the extreme scores. As mentioned earlier, in calculation of the Mean if any one score is extreme and all other scores are near each other, it would give a wrong idea about the average. Let us take an example, again of marks obtained by students of class 10 in a mathematics examination. Let us say the 10 students had scored the following: 55,65,45,35,45,25,50,40,45,100. This extreme score of 100 will affect adversely the Mean ( for instance the M = 50.5. If we remove just that one score of 100, the mean will become 40.5. Thus the score of 100 has made enormous difference to the Mean, that is the measure of central tendency.
As has been pointed out, in the calculation of the Mean, every value is given equal importance. But if one extreme value is present in the series, then the value of mean as seen above becomes misleading.
Similarly let us say that a teacher wants to make comparisons between progress in performance of students of two sections in mathematics. Let us suppose that the average scores obtained by group A in first, second and third term is 60, 20 and 70; let us say the average scores obtained by group B is 45,50,55 in the three terms respectively, then the mean in both the cases is 50. (150 / 3 = 50). However, the lowest and highest scores are 20 and 70 in Group A and 45 and 55 in group B. That is the range is higher (50) in Group A as compared to B (range = 10). This means group B is more homogeneous than Group A in that their performance in the three terms vary far lesser in group B as compared to that in Group A. Group B‘s performance in the three terms appear more consistent than Group A. Hence it is important use the Mean, one of the measures of central tendency with caution, especially when there are extreme scores that may affect the Mean.
1.5.2 The Median
The median is another measure of central tendency.
According to Minium, King & Bear (2001) ―Median is the value that divides the distribution into two halves.‖
According to Garrett (1981) ‗When scores in a continuous series are grouped into frequency distribution, the median by definition is the 50% point in the distribution.‘
If we arrange the items of series in ascending or descending orders of magnitude, the value of the central item in the series is the median. We may say that median is that value of the scores below which 50% of cases lie and also above which 50% of cases lie.
Central Tendencies and Dispersion
scores are as given below for 10 students, 35, 45, 42, 54, 42, 35, 42, 46, 42,^ Con 36, here 42 marks has been obtained by 4 students while all other marks has been obtained by lesser than this number of students. Thus the mode of marks is 42.
1.5.3.1 Properties of the Mode
The mode is easy to obtain. It can easily be identified many times by observation only.
The mode is the only measure of Central tendency which can be used for nominal level variables. For example, there are more Hindus in India than people of any other religion. Here, Hindu religion is referred to as the mode. No other measure of central tendency is appropriate for distribution of religion in India, as one can use mode to describe the most common scores in any distribution.
1.5.3.2 Limitations of the Mode
Mode is not stable from sample to sample. It is affected more by sampling fluctuation. There may be more than two modes for a particular set of scores. For example, if the scores are 4, 9, 5, 6, 5, 4, 8, 7, 3, 10 here 4 and 5 both are mode as both these occur two times whereas all other numbers occur only once.
1.6 LET US SUM UP
A measure of central tendency provides us a single value, which represents the characteristic of the group. The mean, median and mode are most commonly used as measures of central tendency.
Mean can be computed for grouped and ungrouped data. This is the only measure of central tendency based on all the scores in the series.
Median is the score or value of that central item, which divides the series into equal parts. Median is the midpoint of the class intervals.
Mode is the most frequently occurring value.
1.7 UNIT END QUESTIONS
- Given below are statements. Indicate in each case whether statement is true or false. i) Mean is most stable measure of central tendency T/F ii) The median is less affected than the mean by extreme values of observation is a distribution. T/F iii) If the number of observations is even, the median is in the middle of the distribution. T/F iv) Mode is not influenced by extreme values T/F
cies Concept of Central Tendency
Fill in the blanks in the series i) The value of ………………………… is affected by extreme data values. ii) The sign ……………. is used for population mean and sign ……… is used for sample mean. iii) The most frequently occurred value is the series is know as ………. iv) While calculating ………………. every individual item in the data is taken into consideration.
What do you understand by ‗Central tendency‘? Describes the functions of measures of central tendency.
Under what condition median is most suitable than other measures of central tendency?
What are the characteristics of a good measure of central tendency?
Ans. 1 i)T, ii)T, iii)F, iv)T
- i) mean, ii) mue and M iii) mode iv) mean
1.8 GLOSSARY
Measures of Central Tendency : Measure that describes the center of the distribution. The mean, median and mode are three measures of central tendency. Arithmetic Mean : A measure of Central tendency calculated by dividing the sum of observation by the number of observation in the data set. Median : The value of the middle item in the data set arranged in ascending or descending order. It divide the data into two equal parts. Mode : The most occurring value or the value that has the maximum frequency.
1.9 SUGGESTED READINGS
Chaplin, J.P.(1975). Dictionary of Psychology, Alaurel original Drew, B., & Waters, J.(1985).Video games :utilization of a novel strategy to improve perceptual – motor skills in non-institutionalized elderly. Proceeding and Abstract of the Eastern Psychological Association 5,56. English, H.B. & English, A.C (1958). A Comprehensive Dictionary of Psychological and Psycho Analytical Terms , New York : Longmans Green Garrett , H.E. (1981) Statistics in Psychology and Education , Bombay, Vakils, Feffer and Simons Ltd. Guilford, J.P(1965). Fundamental Statistics for Students of Psychology and Education (4th^ ed),New York :McGarw-Hill
