How to determine mode - Quantitative Techniques for management

How mode is calculated by using grouping method ?

Mode is that value of the variate which occurs maximum number of times in a distribution and around which other items are densely distributed. In the words of Croxton and Cowden, “The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated. It may be regarded the most typical of a series of values.” Further, according to A.M. Tuttle, “Mode is the value which has the greatest frequency density in its immediate neighborhood.”

If the frequency distribution is regular, then mode is determined by the value corresponding to maximum frequency. There may be a situation where concentration of observations around a value having maximum frequency is less than the concentration of observations around some other value. In such a situation, mode cannot be determined by the use of maximum frequency criterion. Further, there may be concentration of observations around more than one value of the variable and, accordingly, the distribution is said to be bimodal or multi-modal depending upon whether it is around two or more than two values.

The concept of mode, as a measure of central tendency, is preferable to mean and median when it is desired to know the most typical value, e.g., the most common size of shoes, the most common size of a ready-made garment, the most common size of income, the most common size of pocket expenditure of a college student, the most common size of a family in a locality, the most common duration of cure of viral-fever, the most popular candidate in an election, etc.

Determination of Mode

(a) When data are either in the form of individual observations or in the form of ungrouped frequency distribution

Given individual observations, these are first transformed into an ungrouped frequency distribution. The mode of an ungrouped frequency distribution can be determined in two ways, as given below:

  1. By inspection or
  2. By method of Grouping

(i) By inspection: When a frequency distribution is fairly regular, then mode is often determined by inspection. It is that value of the variate for which frequency is maximum. By a fairly regular frequency distribution we mean that as the values of the variable increase the corresponding frequencies of these values first increase in a gradual manner and reach a peak at certain value and, finally, start declining gradually in, approximately, the same manner as in case of increase.

Example: Compute mode of the following data:

Compute mode

Solution: Writing this in the form of a frequency distribution, we get

frequency distribution

Remarks :

  1. If the frequency of each possible value of the variable is same, there is no mode.
  2. If there are two values having maximum frequency, the distribution is said to be bimodal.

Example : Compute mode of the following distribution

Compute mode

Solution: The given distribution is fairly regular. Therefore, the mode can be determined just by inspection. Since for X = 25 the frequency is maximum, mode = 25.

  • By method of Grouping: This method is used when the frequency distribution is not regular. Let us consider the following example to illustrate this method.

Example : Determine the mode of the following distribution.

Determine the mode

Solution: This distribution is not regular because there is sudden increase in frequency from 20 to 100. Therefore, mode cannot be located by inspection and hence the method of grouping is used. Various steps involved in this method are as follows:

  1. Prepare a table consisting of 6 columns in addition to a column for various values of X.
  2. In the first column, write the frequencies against various values of X as given in the question.
  3. In second column, the sum of frequencies, starting from the top and grouped in twos, are written.
  4. In third column, the sum of frequencies, starting from the second and grouped in twos, is written.
  5. In fourth column, the sum of frequencies, starting from the top and grouped in threes is written.
  6. In fifth column, the sum of frequencies, starting from the second and grouped in threes is written.
  7. In the sixth column, the sum of frequencies, starting from the third and grouped in threes is written.

The highest frequency total in each of the six columns is identified and analyzed to determine mode. We apply this method for determining mode of the above example.

analysis table

Since the value 14 and 15 are both repeated maximum number of times in the analysis table, therefore, mode is ill defined. Mode in this case can be approximately located by the use of the following formula, which will be discussed later, in this chapter.

Mode = 3 Median - 2 mean

Calculation of Median and Mean

Calculation of Median and Mean

Remarks: If the most repeated values, in the above analysis table, were not adjacent, the distribution would have been bi-modal, i.e., having two modes

Example : From the following data regarding weights of 60 students of a class, find modal weight:

modal weight

Solution: Since the distribution is not regular, method of grouping will be used for determination of mode.

grouping table

Since the value 58 has occurred maximum number of times, therefore, mode of the distribution is 58 kgs.

(b) When data are in the form of a grouped frequency distribution

The following steps are involved in the computation of mode from a grouped frequency distribution.

(i) Determination of modal class: It is the class in which mode of the distribution lies. If the distribution is regular, the modal class can be determined by inspection, otherwise, by method of grouping.

Exact location of mode in a modal class (interpolation formula): The exact location of mode, in a modal class, will depend upon the frequencies of the classes immediately preceding and following it. If these frequencies are equal, the mode would lie at the middle of the modal class interval.

However, the position of mode would be to the left or to the right of the middle point depending upon whether the frequency of preceding class is greater or less than the frequency of the class following it. The exact location of mode can be done by the use of interpolation formula, developed below:

Let the modal class be denoted by Lm - Um, where Lm and Um denote its lower and the upper limits respectively. Further, let fm be its frequency and h its width. Also let f1 and f2 be the respective frequencies of the immediately preceding and following classes.

We assume that the width of all the class intervals of the distribution are equal. If these are not equal, make them so by regrouping under the assumption that frequencies in a class are uniformly distributed.

Make a histogram of the frequency distribution with height of each rectangle equal to the frequency of the corresponding class. Only three rectangles, out of the complete histogram, that are necessary for the purpose are shown in the above figure.

Histogram

Histogram

unimodal frequency distribution

Note: The above formulae are applicable only to a unimodal frequency distribution.

Example : The monthly profits (in Rs) of 100 shops are distributed as follows:

monthly profits (in Rs) of 100 shops

Determine the 'modal value' of the distribution graphically and verify the result by calculation.

Solution: Since the distribution is regular, the modal class would be a class having the highest frequency. The modal class, of the given distribution, is 200 - 300.

Graphical Location of Mode

To locate mode we draw a histogram of the given frequency distribution. The mode is located as shown in figure. From the figure, mode = Rs 256.

Determination of Mode by interpolation formula
Since the modal class is 200 - 300, Lm = 200, D1 = 27 - 18 = 9, D2 = 27 - 20 = 7 and h = 100.

Determination of Mode by interpolation formula

method of grouping

Grouping table

method of grouping

Analysis table

Since the two classes, 120 - 130 and 130 - 140, are repeated maximum number of times in the above table, it is not possible to locate modal class even by the method of grouping. However, an approximate value of mode is given by the empirical formula:

Mode = 3 Median - 2 Mean (See § 2.9)

Looking at the cumulative frequency column, given in the question, the median class is
130 - 140. Thus, Lm = 130, C = 46, fm = 21, h = 10.
∴ Md = 130 + 50-46/21 x 10 = 131.9 lbs.

Assuming that the width of the first class is equal to the width of second, we can write

approximate value of mode


Remarks: Another situation, in which we can use the empirical formula, rather than the interpolation formula, is when there is maximum frequency either in the first or in the last class.

Calculation of Mode when either D1 or D2 is negative:

The interpolation formula, for the calculation of mode, is applicable only if both D1 and D2 are positive. If either D1 or D2 is negative, we use an alternative formula that gives only an approximate value of the mode.

We recall that the position of mode, in a modal class, depends upon the frequencies of its preceding and following classes, denoted by f1 and f2 respectively. If f1 = f2, the mode will be at the middle point which can be obtained by adding f2/(f1+f2)*h to the lower limit of the modal class or, equivalently, it can be obtained by subtracting f2/(f1+f2)*h from its upper limit. We may note that f1/(f1+f2)=f2/(f1+f2)= 1/2 when f1 = f2.

Further, if f2 > f1, the mode will lie to the right of the mid-value of modal class and, therefore, the ratio f2/f1 f2 will be greater than 1/2 . Similarly, if f2 < f1, the mode will lie to the left of the mid-value of modal class and, therefore, the ratio f2/f1 f2 will be less than ½ . Thus, we can write an alternative formula for mode as:

alternative formula for mode

Remarks: The above formula gives only an approximate estimate of mode vis-a-vis the interpolation formula.

Example : Calculate mode of the following distribution.

mode

Solution: The mid-values with equal gaps are given, therefore, the corresponding class intervals would be 0 - 10, 10 - 20, 20 - 30, etc.

Since the given frequency distribution is not regular, the modal class will be determined by the method of grouping.

method of grouping

Example : The rate of sales tax as a percentage of sales, paid by 400 shopkeepers of a market during an assessment year ranged from 0 to 25%. The sales tax paid by 18% of them was not greater than 5%. The median rate of sales tax was 10% and 75th percentile rate of sales tax was 15%. If only 8% of the shopkeepers paid sales tax at a rate greater than 20% but not greater than 25%, summarize the information in the form of a frequency distribution taking intervals of 5%. Also find the modal rate of sales tax.

Solution: The above information can be written in the form of the following distribution:

modal rate of sales tax.

Example : The following table gives the incomplete income distribution of 300 workers of a firm, where the frequencies of the classes 3000 - 4000 and 5000 - 6000 are missing. If the mode of the distribution is Rs 4428.57, find the missing frequencies.

missing frequencies

Merits and Demerits of Mode
Merits

  1. Mode is very simple measure of central tendency. Sometimes, just at the series is enough to locate the model value. Because of its simplicity, it s a very popular measure of the central tendency.
  2. Compared top mean, mode is less affected by marginal values in the series. Mode is determined only by the value with highest frequencies.
  3. Mode can be located graphically, with the help of histogram.
  4. Mode is that value which occurs most frequently in the series. Accordingly, mode is the best representative value of the series.
  5. The calculation of mode does not require knowledge of all the items and frequencies of a distribution. In simple series, it is enough if one knows the items with highest frequencies in the distribution.

Demerits

    Mode is an uncertain and vague measure of the central tendency.
    • Unlike mean, mode is not capable of further algebraic treatment.
    • With frequencies of all items are identical, it is difficult to identify the modal value.
    • Calculation of mode involves cumbersome procedure of grouping the data. If the extent of grouping changes there will be a change in the model value.
    • It ignores extreme marginal frequencies. To that extent model value is not a representative value of all the items in a series.

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