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Example – 4. The table below shows the distribution of the scores obtained by 120 shooters in a shooting competition. Using a graph sheet, draw an ogive for the distribution.

Use your ogive to estimate :

1. The lower quartile.
2. The upper quartile.
3. The inter quartile range.
4. The semi inter quartile range.

Solution :- At first we prepare cumulative frequency distribution table.

Now, we draw an ogive.

Fig.- 7 .

As per the graph paper : 

1. Lower Quartile = Q₁ = n/4th observation = 30th observation = 30.
2. Upper Quartile = Q₃ = 3n/4th observation = 90th observation = 57.
3. Inter Quartile Range = Q₃ – Q1= 57 – 30 = 27.
4. Semi Inter Quartile Range = (Q₃ – Q₁) / 2 = 27/2 = 13.5.

Mode :- The observation which occurs most often is called mode. For example mode of the data : 0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 8, 9, 10. is 5, as it has occurred five times.

In a grouped data, the mid value (class mark) of the class having maximum frequency is the mode. For example, let us take a frequency distribution table and find mode.

Here, we see that 30 occurs maximum number of times. Hence mode = 30.

Let us see another example.                                                                                  

Here we see that the class interval 20 – 30 has 26 as maximum frequency. This class is called modal class.

Hence,   Mode = 1/2(20 + 30) = 25.

Formula for calculating mode :- Mode = L + (f_m – f₁) / (2f_m – f₁ – f₂) × i

Where, L = Actual lower limit of the modal class, 
           f_m = frequency of the modal class,
           f₁ = frequency of the class just preceding the modal class,
           f₂ = frequency of the class just succeeding modal class,                                            

i = class size.