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In statistics, how to calculate mode for grouped data ?

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In statistics, how to calculate mode for grouped data ?  [#permalink]

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New post 09 Aug 2018, 11:33
how to calculate mode for grouped data?
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In statistics, how to calculate mode for grouped data ?  [#permalink]

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New post 09 Aug 2018, 22:53
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livedu wrote:
how to calculate mode for grouped data?



Hello livedu


You don't calculate mode in statistics.

There is usually no need for that.

I assume you are asking for mode formula for grouped data, as for data given in intervals, aiming to find mode of modal class, being the class with the largest frequency.

Lets go gradually.


The "mode" is simply the value that occurs most often in one list.

If no number in the list is repeated, then there is no mode for that list.


In general, mean, median, and mode are simply three kinds, and slightly different kinds of "averages" in statistics.

The "mean" will be "average" that you're probably already used to in everyday life. Right technical term is "the arithmetic mean".


We calculate mean by adding up all the numbers on particular list, and then we divide that total number by the number of terms on that list.

The "median" is simply the "middle" value in the particular list of numbers.

To find the median of the list, your numbers have to be listed in numerical order from smallest to largest, so you may have to rewrite your list before you can find the median.


Finally, the "range" of a list of numbers is just the difference between the largest and smallest value on that list.

Lets look at simple example:

Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13


The mean is the usual average, so I'll add and then divide:

(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

Note that the mean, in this case, isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.

The median is the middle value, so first I'll have to rewrite the list in numerical order:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:

13, 13, 13, 13, 14, 14, 16, 18, 21

So the median is 14.

The mode is the number that is repeated more often than any other, so 13 is the mode.

13, 13, 13, 13, 14, 14, 16, 18, 21

The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.


Now different example:


Find the mean, median, mode, and range for the following list of values:

1, 2, 4, 7

The mean is the usual average:

(1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5

The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers. Because of this, the median of the list will be the mean (that is, the usual average) of the middle two values within the list. The middle two numbers are 2 and 4, so:

(2 + 4) ÷ 2 = 6 ÷ 2 = 3

So the median of this list is 3, a value that isn't in the list at all.

The mode is the number that is repeated most often, but all the numbers in this list appear only once, so there is no mode.

The largest value in the list is 7, the smallest is 1, and their difference is 6, so the range is 6.

Finally third case:


Find the mean, median, mode, and range for the following list of values:

8, 9, 10, 10, 10, 11, 11, 11, 12, 13

The mean is the usual average, so I'll add up and then divide:

(8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5-th value; the formula is reminding me, with that "point-five", that I'll need to average the fifth and sixth numbers to find the median. The fifth and sixth numbers are the last 10 and the first 11, so:

(10 + 11) ÷ 2 = 21 ÷ 2 = 10.5

The mode is the number repeated most often. This list has two values that are repeated three times; namely, 10 and 11, each repeated three times.


The largest value is 13 and the smallest is 8, so the range is 13 – 8 = 5.


Remember that the mode can be determined for qualitative data as well as quantitative data, but the mean and the median can only be determined for quantitative data.

Problem


You begin to observe to the color of clothing your employees wear. Your goal is to find out what color is worn most frequently so that you can offer company shirts to your employees.

Monday: Red, Blue, Black, Pink, Green, and Blue

Tuesday: Green, Blue, Pink, White, Blue, and Blue

Wednesday: Orange, White, White, Blue, Blue, and Red

Thursday: Brown, Black, Brown, Blue, White, and Blue

Friday: Blue, Black, Blue, Red, Red, and Pink

What is the mode of the colors above?

The color blue was worn 11 times during the week. All other colors were worn with much less frequency in comparison to the color blue.

The mode is blue.


The mode can also be defined as the most frequently occurring value in a frequency distribution.


Finally, Mode Formula for Grouped Data:

Mode for Grouped Data = L + \(\frac{(fm−f1)h}{2fm−f1−f2}\)

Where,

L = Lower limit of modal class

fm = Frequency of modal class

f1 = Frequency of class preceding the modal class

f2 = Frequency of class succeeding the modal class

h = Size of class interval.


Find the mode of following data

Class- IntervalFrequency
5 - 104
10 - 155
15 - 207
20 - 252


Solution:

Step 1:

Here frequency of class interval 15 - 20 is maximum.
=> 15 - 20 = modal class


Step 2:

L = Lower limit of modal class = 15

fm= Frequency of modal class = 7

f1 = Frequency of class preceding the modal class = 5

f2 = Frequency of class succeeding the modal class = 2

h = Size of class interval = 10 - 5 = 5

Step 3:

Mode = L + \(\frac{(fm−f1)h}{2fm−f1−f2}\)

= 15 + (7−5)∗5 / 2∗7−5−2

= 15 + 107

= 15 + 1.42

= 16.42



I am sure this helps. :cool:
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Re: In statistics, how to calculate mode for grouped data ?  [#permalink]

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New post 09 Aug 2018, 23:05
livedu wrote:
how to calculate mode for grouped data?



20. Descriptive Statistics



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Re: In statistics, how to calculate mode for grouped data ? &nbs [#permalink] 09 Aug 2018, 23:05
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