The following points highlight the three types of measures of central tendency. The types are: 1. Mean 2. Median 3. Mode
Type # 1. Mean:
It is also known as arithmetic mean. It is the ordinary average in Arithmetic.
According to Garrett, “The arithmetic mean, or more simply the mean, is the sum of the separate scores or measures divided by their number.”
Calculation of the Mean when Data are Ungrouped:
In ungrouped data, mean is the sum of separate scores or measures divided by their number. If a man earns Rs. 5, Rs. 6 Rs. 4, Rs. 9 on four successive days his mean of daily wage (Rs. 6) is obtained by dividing the sum of his daily earnings by the number of days he has worked. The formula for the mean (M) of a series of ungrouped measures is: M = ∑X/N (arithmetic mean calculated from ungrouped data).
Where ∑ means the “sum of,”
X stands for a score of other measure,
N is the number of measures in the series.
Calculation of the Mean when Data are Grouped:
When data or measures have been grouped into a frequency distribution, the mean may by calculated by a simple formula:
F stands for the frequency of each class-interval.
X represents the mid-point in each class-interval.
N stands for the total number of frequencies.
The following example is given for calculating the mean from grouped date:
In calculating mean we first of all find the mid-point (x) of various class- intervals. After finding x we find the fx. Column fx is found by multiplying the mid-point (x) of each interval by the number of scores (f) on it; the mean (47.4) is then simply the sum of the fx namely, (2607.5) divided by N (55).
This is known as long method of finding the mean.
Short Method of Finding the Mean:
The long method of finding the mean gives accurate results but often requires the handling of large numbers and entails tedious calculation. Because of this, short method or the “assumed mean” method has been devised for computing the mean.
According to this method we shall find the mean as follows:
Mean = AM + Ci
AM stands for the assumed mean or guessed mean,
C stands for correction in terms of class-interval,
i stands for the length of class-intervals.
Ci stands for fx’/N x 1
Now, we shall calculate the mean by short method from the following data:
The various steps in calculating the mean by the short method may be explained below:
1. In the first column, write the class-intervals.
2. After writing class-intervals, find the mid-points.
3. Then write the frequency.
4. Assume a mean as near the centre of the distribution as possible and preferably on the interval containing the largest frequency.
5. After assuming the mean, find its deviation in terms of class-interval from the assumed mean in units of interval.
6. Multiply each deviation (x’) by its appropriate f, the f opposite to it.
7. Find the algebric sum of the plus and minus of x’ and divide the sum by N, number of cases. This gives C, the correction in units of class interval.
8. Multiply C by the interval length (i) to get Ci, the score correction.
9. Add Ci algebraically to the AM. This will give the true mean.
Advantages of the Mean:
1. It utilises all the items in a group.
2. It is widely understood and easy to calculate.
3. It can be known even when number of items and their aggregate values are known, but details of the different items are not available.
4. It is always definite,
5. It is capable of further algebraic treatment.
6. It is less subject to chance variation. Hence it is more stable measure of central tendency.
Limitations of the Mean:
1. It may not be an actual item in a series.
2. It cannot be computed by merely observing the series, unless the series, is very simple.
3. It is essential to know the actual values of all the items before computing the arithmetic mean, but in the case of median and mode the items on the extreme may be ignored without understanding the values of these measurements.
4. If the number of items in a series is very small, the mean is unduly affected by the extreme items.
When to Use the Mean:
Use the Mean:
1. When the scores are distributed symmetrically around a central point i.e., when the distribution is not badly skewed.
2. When the most accurate measure of the central tendency is desired.
3. When other statistics like standard deviations and co-efficient of correlation are to be computed later. Many statistics are based upon the mean.
Type # 2. Median:
Median is defined as that point on the scale of measurement above which are exactly one half of the cases and below which are the other half.
According to Garrett;
“When ungrouped scores or other measures are arranged in order of size, the median is the midpoint in the series.”
Calculation of the Median when Data are Ungrouped:
This is to say, when ungrouped scores or other measures are arranged in order of size it is exactly the mid-point on the scale.
Two situations arise in the computation of the median from ungrouped data:
(a) When numbers of scores (N) is odd:
To consider the case where N is odd, suppose we have the following scores:
6, 10, 8, 11, 12, 7, 9. First of all we shall arrange the scores in ascending order as;
6,7,8,(9), 10,11,12.
Here we say that the median is 9 because 3 scores are in the left and 3 on the right of it.
(b) When number of scores in even:
Now if we drop the first score of 6 our series contains six scores.
and the median is 9.5. Counting three scores in from the beginning of the series, we complete score 9 (which is 8.5 to 9.5), the upper limit of score 9. In the same way, counting three score in from the end of the series, we move through score 10 (9.5 to 10.5) reaching 9.5, lower limit of score 10.
A formula for finding the median of a series of ungrouped score is Median = {N+1/2} the measure in order of size.
In our second illustration, the median is on the {6+1/2} th or 3.5th score in order of size, that is, 9.5 (upper limit of score 9 or lower limit of score 10).
Calculation of the Median when data are grouped into a frequency distribution: The median in grouped scores is said to be the point of distribution below which and above which lie 50% of the scores.
The formula for finding the median is as under:
L is the lower limit of class-interval in which the median lies.
N/2 is half of the total number of scores.
Fb is the sum of scores below that interval in which the median lies.
fp is the frequency which lies in the interval in which the median lies.
i is the length of the class interval.
Here we give the example of calculating median from the followings grouped data:
Now we shall find out in which class-interval 25 lies. For this we shall count the frequencies from the bottom and will proceed upwards. In the class interval in which 25 lies its lower limit will be taken. Then we shall find out the value of fb and fp and substitute them in the formula.
In the present example:
Steps in Computing the Median:
The following are the steps in computing the median (Mdn) from data tabulated into a frequency distribution:
1. Find the cumulative frequency.
2. Find N/2, that is, one half of the cases in the distribution.
3. On the basis of N/2 find the lower limit of the class-interval upon which the median lies.
4. Find the sum of the scores on all intervals below the median in order to find fb.
5. Put frequency number of scores with the interval upon which the median falls in place of fp.
6. Multiply the result by the size of the class-interval.
7. Add the amount obtained by the calculations into the exact lower limit of the interval which contains the Mdn. This procedure will give the median of the distribution.
Advantages of Median:
1. Its value is rigidly defined.
2. It is easily understood, calculated and interpreted.
3. In certain cases it can be located by inspection.
4. It can be calculated graphically.
5. It is not affected by items on the extreme. One or a few abnormal values may effect greatly the value of the mean, but they do not disturb the median to an appreciable extent. For example, the mean of nine salaries 200,210,230,240, 250,260,270,270 and 270 is Rs. 244.4. If the last salary had been 510, the mean would have been Rs. 271. i.e., Rs. 26.7 more, whereas the median would have remained the same i.e., Rs. 250.
6. It can be applied to cases which cannot precisely be measured quantitatively; such as intelligence, honesty, obedience etc. It is possible, for example, to arrange a set of students according to their intelligence and to find the middle student as representing the class as a whole.
Disadvantages of Median:
1. The median cannot be computed unless the items are arranged in an ascending or descending order. This arranging is sometimes very cumbersome, particularly where the number of items is very large.
2. It is less stable measure than the mean as it is more subject to chance variations.
3. It is not a typical representative of the series, if the items are widely different from one another.
4. It does not prove useful, where items need to be assigned relative importance and weight; because it gives equal weight to all the items.
5. We do not get the aggregate when the median is multiplied with the number of the items.
When to Use the Median:
Use the Median:
1. When the exact mid-point of distribution is wanted – the 50% point.
2. When there are extreme scores or measures which would disproportionately affect the mean. For example, in the scores 8,10,12,14 and 16, both mean and median are 12. But if 16 is replaced by 100, the other scores remaining the same, the median is still 12 but the mean is 28.8.
3. When it is desired that certain scores should influence the central tendency, but all that is known about them is that they are above or below the median.
Type # 3. Mode:
Mode is the score in a given set of data that appears most frequently.
For example, in the series:
4, 6, 16, 12, 16, 18, 24, 16, 30, 32.
16 is the mode because it appears most frequently. When scores are grouped, mode is said to be the middle point of that class-interval which contains for the maximum frequency. According to Guilford, “The mode is strictly defined as the point on the scale of measurement with maximum frequency in a distribution.”
For example in the frequency distribution, the frequency in the class interval 170-174 is maximum. Its midpoint is 172. Therefore this is the mode.
In grouped data, we can also find the mode by the following method:
Mode = 3 Mdn – 2 Mean
(Approximation to the true mode calculated from a frequency distribution)
An example of calculating the modes from the grouped scores is given below:
Firstly, we shall find out the mean and median of this frequency distribution. We shall find out mean by short method.
Merits of Mode:
1. It is very easy to calculate.
2. It can be located by graphic method also.
3. It can often be located by inspection.
4. It can represent series at best because it is the value of that variable which occurs for the maximum times in that series.
5. It is not affected by extreme items. It is often a really typical value.
6. For its determination, it is not essential to know the extreme items, provided the distribution is regular.
Disadvantages of Mode:
1. It is frequently ill-defined and indefinite.
2. It is very delicate measurement. It becomes indeterminate very easily. This is the case with irregular series.
3. It is not amendable to further algebric treatment. It is not based on all the items of a series.
4. The mode multiplied by the number of items does not give the total value of the series.
5. It is not suitable in cases where relative importance of items have to be given due consideration.
6. It may not be fully representative of group in which items of uniform size are comparatively small.
When to Use the Mode:
Use the mode:
1. When a quick and approximate measure of central tendency is wanted.
2. When the most often recurring score is sought.