After reading this term paper you will learn about the three important measures of central tendency used in social research. The measures are: 1. The Arithmetic Mean 2. Median 3. Mode.
1. Term Paper on the Arithmetic Mean:
The arithmetic mean is by far the most common among the averages. It is relatively, easy to calculate, simple to understand and widely used in statistical calculations. The arithmetic mean is defined as the sum of the values of all the items and dividing the total by the number of items.
An example will help us learn how to calculate the arithmetic mean. Let us suppose that eight students receive marks: 54, 58, 60, 62, 70, 72, 75 and 77 respectively, in an examination.
The mean grade score will be:
The procedure in calculating the mean may be expressed in algebraic terms by the following formula:
Where X1 X2, X3, X4…XN are the item-values. N represents number of items.
The above formula can be made more compact by assigning to the arithmetic mean the symbol X̅ (which is read ‘X bar) and using the Σ notation. The symbol Σ is capital sigma.
Let Σ X stand for “The sum of the X’s”
i.e., X1 + X2 + X3 + X4 + … + XN
Then we have:
Arithmetic Mean = X̅ = ΣX/N
Calculating the Mean from the Ungrouped Data:
The above formula can be used to calculate arithmetic mean from ungrouped data.
The following example would illustrate:
The table given below gives the daily wages received by 15 workers in a certain factory. How do we compute the mean daily wages paid to a worker?
If daily wages of worker is represented by ΣX = 258,
we have X̅ = 258/15= 17.2
Mean Daily Wages = 17.2 or Rs. 17.20
This short-cut method of computing mean is based on an important property of the arithmetic mean, i.e., the algebraic sum of the deviations of the various item values from their mean is always equal to zero. This also implies that the sum of the deviations of the various values from some arbitrary value other than the arithmetic mean will not be equal to zero.
This property may be well illustrated with the following example:
This phenomenon shall be so for any series of values. Thus, we can assume an arbitrary mean to calculate the deviations of their values from this assumed mean. If the sum of deviations is divided by the number of items and added to the assumed mean, we shall get the actual arithmetic mean.
Arithmetic Mean X̅ = a + Σd’/N
Where X̅ is the arithmetic mean,
α is the assumed arithmetic mean.
Σd’ is the sum of deviations from the assumed mean.
N is the number of item values.
If we compute mean daily wages paid to workers from the data given in example by this short-cut method it will give us exactly the same answer as we get by the direct method.
This is illustrated below:
Arithmetic Mean or X̅ = α + Σd’/N
= 20 + (- 42/15)
Mean daily wages = Rs. 17.20
Calculation of Arithmetic Mean from Grouped Data:
Discrete Series: Direct Method: In discrete series the values of the variables are multiplied by their respective frequencies and the sum of the product so obtained is divided by the sum of the frequencies. The resulting quotient is the arithmetic mean of the series.
where X1, X2, X3, X4 … XN are the values, and f1, f2, f3, f4 … fN their respective frequencies.
The following table shows gains in weight of 65 children during a specified period. Calculate mean weight gained by children.
X̅ = ΣXf/N = 346/65 = 5.32
Mean weight gained = 5.32 Kg. (approx.).
In this method the deviation of each item value from the assumed mean is found out and is multiplied by its corresponding frequency. The algebraic sum of the product is divided by the total frequency (N) and added to the assumed mean.
Symbolically X̅ =α + Σf.dx/N
Where ‘α’ stands for the assumed mean; Σf. dx for the sum of deviations from the assumed mean; and N for the total number of frequencies.
Calculation of arithmetic mean Σf .dx from the data given in example 2.
X̅ = α+ Σf.dX /N =5+21/65= 5+0.32 =5.32 (Approximately).
Mean weight = 5.32 Kg. (Approximately).
Direct Method: In calculating arithmetic mean of a continuous series, we take the mid-value of each class as representative of that class (since it is presumed that the frequencies of that class are concern on mid-point) multiply the various mid-values by their corresponding frequencies and sum of the products is divided by sum of the frequencies.
The following example will illustrate the method:
Arithmetic Mean = Rs. 187.16 (Approximately)
Arithmetic Mean (X̅) = α + Σfd/N x 1
Where ‘α’ stands for assumed mean Σfd’ for sum of total deviations, W for total number of frequencies and T for class interval. Now substituting the values in the formula from the table.
= 180+ 136/380 x20
= 180 + 7.16
Arithmetic Mean = Rs.187.16 (Approximately).
It is necessary to note that while determining the class interval when the class interval is exclusive (say 15-20, 20-25, 25-30) the method of finding class interval (i) will be as follows:
Class interval (i) Upper limit and lower limit = 20 – 15 = 5.
If, however, the class-interval is inclusive say (15-19, 20-24, 25-29) the apparent limits are not real limits and hence before finding the class-interval, real limits would need to be determined.
In determining the class interval of inclusive series it is assumed that figures have been rounded to the nearest unit given. For example, if the classes are written as 10-14, and 15-19, the lower and upper limits would be considered as 9.5-14.5 and 14.5-19.5, respectively.
Steps in the Short-cut Method:
The procedure for computing arithmetic mean may be schematically expressed as follows:
2. Term Paper on the Median:
The median is another simple measure of central tendency. We sometimes want to ascertain the position of the middle item, when data have been arranged, or we want to divide a group of students into quartiles by locating the individuals who have exactly 25 per cent of the class below them, or exactly 75 per cent below them etc.
These measures are known as positional averages. We define the median as the size of the middle item when the items are arrayed in ascending or descending order of magnitude. This means that median divides the series in such a manner that there are as many items above (or larger than the middle one) as there are below (or smaller than) it.
Location of Median in a Series of Ungrouped Data:
To locate the median in ungrouped data-set, first the data are arrayed in ascending or descending order. If the number of items is odd the middle item of the array is the median. If X is even, the median is the average of the two middle items. Symbolically, the median is:
Median (X) = The (N+1/2)th item in the series arrayed in order of magnitude.
Example: The following table gives the daily wages received by 15 workers in a certain factory. Compute the median wages of the data-set.
Median (X) = Size of the (N+1)/2th item in the data-set arrayed in order
= N+1/2 = 15/+1/2 =16/2 =8
Size of 8th item = 17
Median wage of the series = Rs.17.00.
If the number of items in the series is even, the median is taken as the arithmetic mean of the middle items.
If one item is removed from the above example, the median would be:
X = the size of (14+1/2) th item in the data-set.
= the size of 7.5th item.
Since the median is the 7.5th item in the arrayed series, we need to average 7th and 8th items. The seventh item in Table 18.23 is 16, and the eighth is 17.
Median (X)= the size of 7.5th item 16+17/2 = 16.5
The median wages of 14 workers is Rs.16.50
Location of Median in Discrete Series: In a discrete series the frequencies are cumulated and then the value of the middle item is located.
The following example will illustrate the steps:
X = Gain in weight of N/2th item.
= Gain in weight of 65/2 th item.
= Gain in weight of 32.5 th item.
Median (X) = 5 Kg.
(It is clear from the above Table that the value of items 23rd to 34th is 5. Thus, the value of 32.5th item is also 5).
Determination of Median in a Continuous Series:
In a continuous series we do not know every observation. Instead, we have record of the frequencies with which the observations appear in each of the class intervals as in the following Table. Nevertheless, we can compute the median by determining which class interval contains the median.
In regard to data given in Table 18.26 the median value is the value on either side of which N/2 or 380/2 or 190 items lie. Now, the problem is to find the class interval containing the 190th item. The cumulative frequency for the first three classes is only 105.
But when we move to the fourth class interval95 items are added to 105 for total of 200. Therefore, the 190th item must be located in this fourth class interval (the interval from Rs.170 and Rs.190).
The median class (Rs.170 – Rs.190) for the series contains 95 items. For the purpose of determining the point, which has 190 items on each side, we assume that these 95 items are evenly spaced over the entire class interval 170-190. Therefore, we can interpolate and find the value for 190th item. First, we determine that the 190th item is the 95th item in the median class.
? 190 – 105 = 85
Then we can calculate the width of the 95 equal steps from Rs.170 to Rs.190 as follows:
190-170/95 = 0.21053 (Approximately)
The value of 85th item is 0.21053 x 85 = 17.89. If this value (17.89) is added to the lower limit of the median class, we get 170 + 17.89 = 187.89. This will be the median of the series.
This can be put in the form of a formula.
Where X = Median
L = Lower limit of the class in which median lies
N = Total number of items
C = Cumulative frequency of the class prior to the median class
F – Frequency of the median class
i = Class interval of the median class
Calculation of Median by Formula
The procedure for computing median may be schematically expressed as under:
3. Term Paper on the Mode:
Another measure which is sometimes used to describe the central tendency of a set of data is the mode. It is defined as the value that is repeated most often in the data-set. In the following series of values: 71, 73, 74, 75, 78, 80 and 82, the mode is 75, because 75 occurs more often than any other value (three times).
In grouped data the mode is located in the class where the frequency is the greatest. The mode is more useful when there are a larger number of cases and when data have been grouped. In practice, it is found that a set of grouped data often have more than one mode.
Frequency distributions which have one mode are referred to as uni-modal. Those with two modes and three modes are described as bimodal and tri-modal (more commonly multimodal) distributions.
Calculation of Mode in Discrete Series:
In a discrete series the value against which the frequency is the largest would be the modal value. If we look at table 18.25 we will find that 12 is the largest frequency. Thus, 5 being the value against which the frequency is the largest, is the mode of this series.
If however, there are more than one value round which figures concentrate it becomes difficult to determine the value of mode. In such cases attempts are made to find out the point of maximum concentration with the help of grouping method.
The procedure is as follows:
(i) First the frequencies are added in ‘two’s, in two ways:
(a) By adding frequencies of item numbers 1 and 2; 3 and 4: and 5 and 6; and so on, and (b) by adding frequencies of item numbers 2 and 3, and 5, 6 and 7 and so on.
(ii) Then the frequencies are added in ‘three’s: This can be done in three ways:
(a) By adding frequencies of item numbers 1, 2 and 3, 4 and 5 and 6 and 7, 8 and 9 and so on,
(b) By adding frequencies of item numbers 2 and 3 and 4, 5, 6 and 7, 8, 9 and 10; and so on, and
(c) By adding frequencies corresponding to item numbers 3, 4 and 5, 6, 7 and 8, 9, 10 and 11 and so on.
If necessary, grouping of frequencies can be done in ‘four’s and ‘five’s also. After grouping, the size of items maximum frequencies are circled. The item value contains
corresponding the maximum frequency, the largest number of times is the mode of the series. This is shown is tables 18.28 and 18.29.
To find out the point of maximum concentration, the data may be arranged in the form of table as follows:
Since, value 6 kgs. occurs the largest number of times it is the modal value. Thus, the mode of the series is 6 kg.
Determination of Mode in a Continuous Series:
In the case of continuous series, the determination of mode involves two steps. First, the class of maximum concentration is located by the process of grouping. After this, the value of mode is interpolated by the use of the following formula:
Here, x stands for the mode, L is the lower limit of the modal class, f0 stands for the frequencies of the preceding class, f1 for the frequencies of the modal class, f2 for the frequencies of the succeeding class and i stands for the class interval or the modal class.
Illustration: Determine mode of the following distribution:
Therefore, 300-400 group is the modal group. Using the formula of interpolation, viz.:
X = 11+ f1-f0 / 2f1 –f0-f2 x i, we get the modal value.
The three measures of central tendency discussed at length above not only make different assumptions about the nature of the data but provide also somewhat different kinds of information. The arithmetic mean may be thought of as the point on the scale around which the cases balance, that is, one case at once extreme of the distribution may be counter-balanced by one or more cases at the other extreme of the distribution.
If cases are not symmetrically distributed around this point, the arithmetic mean may lead one to draw inferences about the group which could be grossly unrealistic, i.e., it may give out a distorted picture of the group in respect of given characteristics.
This does not, however, mean that the knowledge of the mean (say, of the mean income of a group) is a pointless information. Whether or not the knowledge of the mean is useful will depend on what one proposes to do with it.
The median also implies a concept of balance, but since it takes account only of the ordinal positions of scores rather than their absolute values, a person with a very high income may be counter-balanced by a pauper. But in the case of the arithmetic mean it takes a very large number of cases of low income to counter-balance one case of a person who has a very high income.
The use of the mode as a measure of central tendency does away with the idea of a point of balance, since, it is possible to have more than one mode in a distribution. For example, we may find two equally large groups of students, one studying between 1 and 2 hours and another studying between 3 and 4 hours. In this distribution, there will thus be two modes, i.e., the distribution is bimodal.
These may also be multi-modal distributions. In case there are, it is perhaps misleading to speak of central tendency. It is also possible for a distribution of scores to have no mode at all, i.e., every score may be occurring about as frequently as every other.
In this case too, it is misleading to speak of central tendency, a term suggesting that the cases tend to cluster around some point more or less in the middle of the distribution. In such instance, the arithmetic mean and the median might be useful in bringing out salient aspects of the distributions which they are trying to depict.
In social research, one may also wish to know certain other aspects of how individuals or cases in the sample or ‘population’ with respect to the variable being measured (e.g., hours of study ‘intelligence quotient, etc.’) are distributed.
One may want to know, for instance, whether the number of students who study for 2.5 hours on an average is about the same as the number that studies for 4.5 hours on an average or whether relatively more students go to cinema three times a month, while few do not go at all and relatively fewer still go to the cinema six or more times a month.
If one plotted the figures on a graph indicating the frequency of movie-going on the horizontal axis and the number of people corresponding to each frequency on the vertical axis what would be the shape of the resulting graph.
Let us illustrate this on the basis of the above data (few students at the ends, i.e., those not going to films at all and those visiting films four or more times per year and a large number of students visiting films about twenty times a month.)
The shape of the above distribution is like a bell (known technically as bell-shaped or ‘normal’ distribution or symmetrical distribution). With different kinds of the data characterized by a piling of cases on one side rather than on the other, the curve would be asymmetrical. The figure given below represents a distribution that is asymmetrical.
With yet another kind of data characterized by pilling up of cases at two (or more) points along the scale with relatively few cases in between, we would get a bi-modal (or multi-modal) distribution. The following figure illustrates a bi-modal distribution.
It should be noted that knowing the shape of the distribution curve is fundamental to the use of efficient statistical methods, since the more efficient methods make specific assumptions about the nature of the distribution curve. It is customary to assume that the distribution curve for any variable is normal but in reality, this may not be so.
If it is not, one may see whether other known distribution curves fit the data or whether one can transform the raw data by mathematical manipulation to a known distribution curve. Any standard book of statistics would help clarification on this point.