Central Tendency Extended
1. Median Introduction And Definition
Median is the middle most items in a statistical distribution. Median is the value of the middle item of a series arranged in ascending or descending order of magnitude. It is that positional value which reflects only middle 50% of the observations. It excludes top 25% and bottom 25% of the observations. Unlike arithmetic average median does not take into account the values of all items in a series. It is for this reason that median is called a 'positional average. Its value is the value of the middle item irrespective of all other values. In case of arithmetic average values, of all items are taken into account and that is why, it is a 'mathematical average’. L.R. Connor defines median as "that value of the variable which divides the group into two equal parts, one part comprising all values greater, and the other, all values less than median." The method to calculate the median is very simple. In case of an observation called marks secured by 5 students in an examination are as 45, 67, 78, 89, 90. Median lies in the class n+1/2 th item in the series. N+1/2 th item = 3 rd item. Before that the series need to be sorted either in ascending or descending order.
2. Computation of Mode
Modal value is the highest observation which has more frequency in comparison to other frequencies. It appears from the definitions given above that it must be very easy to calculate the mode of a series. In fact it is not always so. The most satisfactory method of calculating mode is that of "curve fitting’' which is an extremely difficult process. In ordinary practice, however, mode is estimated by easier methods which are comparatively very much less accurate than the method of curve fitting. These methods are no doubt very simple and easy.
3. Determination Of Mode In Continuous Series
In a continuous series the determination of mode involves two steps. First, by the process of grouping, the class in which there is maximum concentration has to be located. After this the value of mode is interpolated by the use of a formula. It should be remembered that mode does not always give satisfactory results in a continuous series. If the size of the class-interval is changed the modal class also changes in many cases. Suppose, for example, the magnitude of class-intervals is 10 and mode lies in, say, 30—40 group. If this series is regrouped in class-intervals having magnitude of only 5, it is quite likely that the mode may lie in, say, 45—50 group. It would depend on the distribution of items in various class intervals. For determining mode in a continuous series, the class-intervals should not be very big in size, but if the size of the class-intervals is very small the frequencies also become very small, the distribution becomes irregular, and the determination of mode becomes very difficult. The series may become bi-modal or multi-modal. It has already been said that the mode is affected by the frequencies of the neighboring classes. The formulae for the interpolation of mode are based on this very assumption. If the frequency of the preceding class is greater than the frequency of the succeeding class, mode would be nearer the lower limit of the class-interval and if the frequency of the succeeding class is more than the frequency of the preceding class mode would be nearer the upper limit. To study this, the proportions of frequencies in the preceding and succeeding classes to the total frequencies in these two classes are found out.
If f1 stands for the frequencies of the preceding class and f2 for the frequencies of the succeeding class these proportions would be :
f0 / f0+ f2 and f2 / f0 - f2
These proportions are multiplied by the magnitude of the class interval and mode is calculated in any of the following two ways either
by adding f0/(f0+f2 ) ×(l2—l1) to the lower limit of the modal class or
by deducting f0/(f0-f2 ) X (l2—l1) from the upper limit of the modal class. Thus if Z stands for the mode,
Z = l1+f2 / f0 + f2 ) × I Z = l2+f0 / f0 + f2 ) × I I = (l2-l1)
Mode is also calculated by taking into account (i) the proportion of difference between the frequency of the modal class and the frequency of the preceding class and (ii) the proportion of difference between the modal frequency and the frequency of the succeeding class. Thus if f1 stands for the frequency of the modal class, and if we take into account the lower limit of the modal class, the proportion of the difference (f1-f0) is added to it and if we take into account the upper limit, the proportion of the difference (f1-f2) is deducted from it.
Thus,
The two sets of formulae given above would give different values of mode as they are based on different assumptions. In the first case we take into account only the frequencies of the preceding and succeeding classes whereas in the second case (i) difference of the modal frequency and the preceding frequency, and (ii) the difference of the modal frequency and the succeeding frequency, are taken into account.The first set of formulae are supposed to be better than the second set and usually mode is interpolated by starting with the lower limit. As such we shall be making use of the following formula in the determination of mode in a continuous series :
Z = l1 + f1 - fo / 2f1 - fo-f2 × (l1-l2)
Some authors have expressed this very formula in a different form as follows :
Z = l1 + Δ1 / Δ1 + Δ2 × i
Where Δ1 is the difference between the frequency of the modal class and the frequency of the pre-modal class (f1—f0) and Δ2 is the difference between the frequency of the modal class and the frequency of the post-modal class (f1—f2) ignoring + or — signs.
Thus. Δ1+ Δ2 = f1—f0+ f1—f2 =2f1- f0- f2. In this formula i stands for the magnitude of the modal Class (l2—l1). Thus the two forms of the formula are the same.
4. Mode
Mode is the most common item of a series. Generally it is the value which occurs the largest number of times in a series. People indicate that mode is a value around which there is the greatest concentration of values. It may not necessarily be the value which occurs the largest number of times in a series, as in some eases the point of maximum concentration may be around some other value. In some cases there may be more than one point of concentration of values and the series may be bi-model or multi-modal. We shall discuss these cases later.The word Mode is derived from the French word {la mode) which mean fashion or the most popular phenomenon. Mode, thus is the most popular item of a series around which there is the highest frequency density. When we speak of the 'average student', average collar size', 'average size of a shoe', we are referring to mode.
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