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Calculation of Mode in Continuous Series

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The ‘word’ mode gets its roots from nan aged French connection ‘La Mode’, which intends astir celebrated phenomenon, and from nan Latin connection ‘Modus’, meaning measurements, quantity, measurement aliases manner. However, successful English, nan literal meaning of mode is ‘the astir predominant worth successful a group of values’. Mode, therefore, successful statistics, refers to nan adaptable that occurs astir of nan clip successful nan fixed series. In elemental words, a mode is simply a adaptable that repeats itself astir often successful a fixed bid of variables (say X). We tin find nan mode successful 2 series; viz., individual and discrete series. 

Mode is denoted arsenic ‘Z‘.

Continuous Series:

A discrete bid cannot return immoderate worth successful an interval; therefore, successful cases wherever it is basal to correspond continuous variables pinch a scope of values of different items of a fixed data, Continuous Series is used. In this series, nan measurements are only approximations and these approximations are expressed successful nan shape of people intervals. The classes are formed from opening to end, without immoderate breaks. Mode successful a continuous bid belongs to a circumstantial people aliases group that is known arsenic nan modal class. 

Methods of Calculating Mode successful Continuous Series

There are 2 methods of calculating Mode successful Continuous Series:

  • Observation Method aliases Inspection Method
  • Grouping Method

1. Observation Method

The study method tin beryllium utilized to place mode if nan frequencies are uniform, homogeneous, and person only 1 maximum frequency. When nan frequencies of a continuous bid emergence and autumn successful immoderate systematic order, nan modal people tin beryllium wished simply by inspection of the series.

Steps to cipher mode utilizing Observation Method successful lawsuit of Continuous Series

Step 1: Identify nan modal class, which intends nan people pinch nan highest frequency.

Step 2: The nonstop worth of nan mode tin beryllium calculated by using the pursuing formula:

 M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

Where,  

Mo = Mode

l1 = Lower limit of modal class

f1 = Frequency of modal class

f0 = Frequency of people preceding nan modal class

f2 = Frequency of nan people succeeding nan modal class

i = Class interval of nan modal class

The look tin besides beryllium expressed as:

 M_o=l_{1}+\frac{f_1-f_0}{(f_1-f_0)+(f_1-f_2)}\times{i}

Note:

If nan wave of nan pre-modal people aliases post-modal people is higher than nan modal class; i.e., if (f1 – f2) is antagonistic aliases (2f1 – f0 – f2) is zero, past nan look for calculating mode will be:

M_o=l_1+\frac{|f_1-f_0|}{|f_1-f_0|+|f_1-f_2|}\times{i}

The look and nan meanings of nan symbols are same, nan only quality is that we person taken absolute values successful this look aft ignoring antagonistic signs.

Example: Find retired nan mode of nan pursuing series.

Data Table

Solution: By looking astatine nan data, it is evident that nan modal people is 20-30 because nan wave of this people is nan maximum; i.e., 15.

Mode Table

 M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

Where, l1 = 20, f1 = 15, f2 = 8, f0 = 10, and one =10

M_o=20+\frac{15-10}{2(15)-10-8}\times{10}

M_o=20+\frac{5}{30-18}\times{10}

M_o=20+\frac{50}{12}

Mo = 20 + 4.16

Mode (Z) = 24.16

2. Grouping Method

The inspection method tin beryllium utilized if nan frequencies are uniform, homogeneous, and person only 1 maximum frequency. But successful lawsuit of irregularity, Grouping Method is preferable. In nan grouping method, frequencies are grouped to get a unsocial shape and 2 tables are prepared to find nan mode, viz., The Grouping Table and The Analysis Table.

Calculation of  Mode utilizing Grouping Method successful lawsuit of Continuous Series

 Grouping Table

 Prepare 6 columns successful summation to nan file of Class Interval (X) and past shape groups according to nan pursuing instructions: 

  1. Firstly, return nan fixed frequencies successful file 1.
  2. Then return nan sum of frequencies successful two(s) successful file 2. 
  3. Now successful file 3, return nan sum of frequencies successful two(s), starting from nan 2nd worth of nan fixed frequencies. 
  4. Take nan sum of frequencies successful three(s) successful file 4.
  5. In file 5, return nan sum of frequencies successful three(s), starting from nan 2nd worth of nan fixed frequencies.
  6. Lastly, successful file 6, return nan sum of frequencies successful three(s), starting from nan 3rd worth of nan fixed frequencies.

After preparing each six columns, underline, circle, aliases item nan maximum wave (maximum total) of each column.

 Analysis Table

  1. First, hole different array showing each six columns vertically and each nan fixed values of nan Class Interval (X) horizontally.
  2. Now according to nan Grouping Table, people (✓) nether that people interval which is portion of nan maximum full of nan file nether consideration.
  3. Repeat nan process for each of nan columns and people (✓) nether nan concerned people interval.
  4. Now count nan people (✓) of each file successful nan Analysis Table.
  5. The Class Interval pinch a maximum number of ticks (✓) is wished to beryllium nan Modal Class for nan fixed series, i.e., nan modal people forming astir of nan highest grouped frequencies is wished arsenic nan modal class, which is past utilized successful nan look to find nan mode.

Steps successful Grouping Method

Step 1: First of all, find nan modal people by nan process of grouping arsenic stated above.

Step 2: Calculate nan nonstop worth of nan mode by using the pursuing formula:

 M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

Example: Find retired nan mode of nan pursuing bid utilizing nan Grouping Method.

Data Table

Solution: The modal people is not clear upon person investigation. Although nan largest wave (26) is successful nan 10-15 class, nan top attraction of items is astir nan 20–25 people (with a wave of 21). As a result, an study array and grouping array are created to cipher nan mode.

Grouping Table

Grouping Table

  • Column (I) shows nan wave of nan fixed series, and 26 is marked arsenic nan highest value.
  • Column (II) shows nan sum of frequencies successful two(s), i.e., [5+12=17, 26+13=39, 24+21=45, and 11+4=15]. In this column, nan highest worth marked is 45.
  • Column (III) shows nan sum of frequencies successful two(s), starting from nan 2nd worth of fixed frequencies, i.e., [12+26=38, 13+24-37, and 21+11=32]. The highest worth marked is 38.
  • Column (IV) shows nan sum of frequencies successful three(s), i.e., [5+12+26=43 and 13+24+21=58]. In this column, nan highest worth marked is 58.
  • Column (V) shows nan sum of frequencies successful three(s), starting from nan 2nd worth of fixed frequencies, i.e., [12+26+13 and 24+21+11]. The highest worth marked is 56.
  • Column (VI) shows nan sum of frequencies successful three(s), starting from nan 3rd worth of fixed frequencies, i.e., [26+13+24=63 and 21+11+4=36]. The highest worth marked is 63.        

Analysis Table

Analysis Table

  • 26 is nan highest worth successful nan first column, corresponding to nan people interval 10-15. So, we person marked (✓) nether 10-15 successful nan study table. 
  • In nan 2nd column, nan highest worth marked is 45, and it is nan sum of 24 and 21, i.e., corresponding to nan people intervals 20-25 and 25-30. Therefore, we person marked (✓) nether 20-25 and 25-30.
  • In nan 3rd column, nan highest worth marked is 38, and it is nan sum of 12 and 26, i.e., corresponding to people intervals 5-10 and 10-15. Therefore, we person marked (✓) nether 5-10 and 10-15.
  • In nan 4th column, nan highest worth marked is 58, and it is nan sum of 13, 24, and 21, i.e., corresponding to people intervals 15-20, 20-25, and 25-30. Therefore, we person marked (✓) nether 15-20, 20-25, and 25-30.
  • In nan 5th column, nan highest worth marked is 56, and it is nan sum of 24, 21, and 11, i.e., corresponding to people intervals 20-25, 25-30, and 30-35. Therefore, we person marked (✓) nether 20-25, 25-30, and 30-35.
  • In nan sixth column, nan highest worth marked is 63, and it is nan sum of 26, 13, and 24, i.e., corresponding to people intervals 10-15, 15-20, and 20-25. Therefore, we person marked (✓) nether 10-15, 15-20, and 20-25.

According to nan Analysis Table, nan highest number of ticks (✓) is against nan people interval 20-25; therefore, nan modal people of nan fixed bid is 20-25.

Now, pinch nan thief of nan pursuing formula, nan mode will be:

 M_o=l_{1}+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}

Where, l1 = 20, f1 = 24, f2 = 21, f0 = 13, and one = 5

M_o=20+\frac{24-13}{2(24)-13-21}\times{5}

M_o=20+\frac{11}{48-34}\times{5}

M_o=20+\frac{55}{14}

Mo = 20 + 3.92

Mode (Z) = 23.92


Editor: Naga



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