The mode is defined to be size of the variable which occurs most frequently or the point of maximum frequency or the point of maximum frequency or the point of greatest density. It is also an important measure of central tendency. For example, we have given the data, that is, 19, 21, 20, 19, 19, 19, 25, in this case, 19 is the mode value which is occurring very frequently. It is denoted by ‘Z’.
According to Kenny and Keeping:
“The value of the variable which occurs most frequently in a distribution is called the mode”.
Z= L+ f1 – f0 /2 f1– f0– f2 ×i
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In Section Method:
In this method, we just inspect the items by observations, and if any value is occurring very frequently, it will be treated as ‘mode’.
Example 1:
X: 3, 1. 9, 2, 8, 5, 8
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Mode = 8 because this value is more repeated.
Conversion of Individual Series into Discrete Series:
When there is large number of items in a given series then convert the individual series into discrete series firstly. If any value has highest frequency that value is known as mode.
Discrete Series:
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In discrete series we use both methods, that is, inspection method and grouping method.
Inspection Method:
In this method, we inspect the whole series and if any value has highest frequency then this value will be taken as ‘mode’.
Example 2:
Discreet Series (Direct method):
Solution:
The highest frequency is 8 therefore, the corresponding value 6 is the value of mode. Continuous series (Direct method)
The formula of this series is:
L1 = lower value of class limit
f1 = highest value of frequency
f0 = preceding the value of highest frequency
f2 = succeeding the value of highest frequency
i = difference between two variables.
Calculated in special circumstances when 2f1 – f0 – f2 – 0.
Under such cases the above formula with all the symbols used as explained can be determined as:
Example 3:
Find the value of mode of the following data:
Find the value of mode.
Solution:
The highest value of frequency is 10 therefore mode lies between 16-24
Note:
When in the question same value is repeated more than one time or one value of upper or lower of the highest frequency is < 3 then in that situation we apply the grouping method. Otherwise in discrete as well as continuous series, we apply, direct method.
Grouping Method:
Generally, through inspection of a given series, we can find the ‘mode’ value very easily because in it, if any value is coming very frequently and if any value has greatest frequency then it will be treated as ‘mode’ but sometimes, it is not possible to locate the value.
This problem arises when the difference between the maximum frequency preceding or succeeding it is very small and items are heavily concentrated on either side. This problem can be solved through grouping method. The mode value is determined by two tables, that is, grouping table and analysis table.
Example 4:
Calculate the mode for the following distribution:
Solution:
Here, the largest frequency is 72. It lies in the class 21-28, so the model class is 21-28 and the lower limit of the model class is 21.
Thus,
Grouping Table:
Grouping table has the following procedure to find out the modal value:
1. In the first, put the maximum frequency in circle.
2. In the second, group the first two frequencies by taking the total of them.
3. In the third, leave first one and group the remaining two.
4. In the fourth first three frequencies are grouped.
5. In the fifth, first frequency is left and the remaining group in three.
6. In the sixth, leave the first two frequencies and group the other frequencies in three. This column will be used as the base for the preparation of analysis table.
7. Make circle or mark the highest number in all six columns.
Analysis Table:
After making the grouping table, analysis table is prepared. It involves the following steps:
1. Place the column number on the left side vertically.
2. Place the probable values of X horizontally.
3. Now, enter the highest frequencies which are marked in grouping table as circle.
Graphical Presentation of Mode:
Graphical presentation of mode is also possible like other positional averages that is, median, quartile, decides, percentiles.
It has following procedure:
1. First of all, draw a histogram on the basis of given data.
2. The rectangle which has maximum height will show the model value.
3. Then, join the left-hand corner of the highest rectangle with the left-hand corner of the-next rectangle and the right-hand corner of the highest rectangle is joined with the right-hand corner of the preceding rectangle.
4. At the point on which both lines intersect, draw a perpendicular from this point on X-axis. It will show the model value.
Example 5:
Draw a histogram for the following distribution and find the model wage and check the value by direct calculation:
Solution:
It is clear from the histogram that the model value is direct calculation.
Mode lies in the class 25-30, that is, 27.85
The sight difference in the two answers is due to the difficulty in reading from the graph.
Mode can also be determined from a frequency polygon in which case a perpendicular is drawn on the base from the apex of the polygon and the point where it meets the base gives the model value.
However, graphic method of determining of mode can be used only where there is one class containing the highest frequency. If two or more classes have the same highest frequency, mode cannot be determined graphically.
Merits and Demerits of Mode Merits:
1. It is very simple to understand and easy to calculate because it is a positional average,
2. This is based on quality rather than quantity.
3. It is least effected by the extreme values.
4. Where there is a large concentration of items around the value, that value is the good representative of the items.
5. It is possible graphically to show the model value.
Demerits:
1. It is not a suitable measure of central tendency where the number of items is very small
2. It has no further mathematical applicability.
3. If we have given the data about more than two series, then it is not possible to calculate model value.
4. It is not possible to find out the sum of the items by multiplying with the model value the number or items in this measure of central.
Relationship among Mean, Median, Mode:
Mean, median and mode are the important measures of central tendency. There is a very close relationship between the three of them.
1. In a perfect mathematical distribution, the mean, median and mode will be the same.
Symbolically, X̅ = M = Z
2. In a moderate distribution, the relationship between mean, median and mode can be shown through following points:
(i) The distribution is positively skewed when it shift to right.
Symbolically; X̅ >M>Z
Positively skewed distribution
(ii) The distribution is negatively skewed when it shifts to left Symbolically, X < M< Z
