Question 1:
Find the mode and median of the data: 13, 16, 12, 14, 19, 12, 14, 13, 14
By using the empirical relation also find the mean.
Arranging the data in ascending order such that same numbers are put together, we get:
12,12,13,13, 14,14,14, 16, 19
Here, n = 9.
∴ Median = Value of observation = Value of the 5^{th} observation = 14.
Here, 14 occurs the maximum number of times, i.e., three times. Therefore, 14 is the mode of the data.
Now,
Mode = 3 Median  2 Mean
⇒ 14 = 3 x 14  2 Mean
⇒2 Mean = 42  14 = 28
⇒ Mean = 28 ÷ 2 = 14.
Find the median and mode of the data: 35, 32, 35, 42, 38, 32, 34
Arranging the data in ascending order such that same numbers are put together, we get:
32, 32, 34,35,35, 38,42.
Here, n = 7
∴ Median = Value of observation = Value of the 4^{th} observation = 35.
Here, 32 and 35, both occur twice. Therefore, 32 and 35 are the two modes.
Find the mode of the data: 2, 6, 5, 3, 0, 3, 4, 3, 2, 4, 5, 2, 4
Arranging the data in ascending order such that same values are put together, we get:
0, 2, 2, 2, 3, 3,3,4,4,4,5,5,6.
Here, 2,3 and 4 occur three times each. Therefore, 2 ,3 and 4 are the three modes.
Alternate Solution
Arranging the data in the form of a frequency table, we have:
Values  Tally Bars  Frequency 
0  ∣  1 
2  ∣∣∣  3 
3  ∣∣∣  3 
4  ∣∣∣  3 
5  ∣∣  2 
6  ∣  1 
Total  13 
Clearly, the values 2,3 and 4 occur the maximum number of times, i.e., three times.
Hence, the mode is 2,3 and 4.
The runs scored in a cricket match by 11 players are as follows:
6, 15, 120, 50, 100, 80, 10, 15, 8, 10, 10
Find the mean, mode and median of this data.
Answer 4:
Arranging the data in ascending order such that same values are put together, we get:
6,8,10,10,10,15,15,50,80,100,120.
Here, n = 11
∴ Median = Value of observation = Value of the 6^{th} observation = 15.
Here, 10 occurs three times. Therefore, 10 is the mode of the given data.
Now,
Mode = 3 Median  2 Mean
⇒ 10 = 3 x 15  2 Mean
⇒2 Mean = 45  10 = 35
⇒ Mean = 35 ÷ 2 = 17.5.
Find the mode of the following data:
12, 14, 16, 12, 14, 14, 16, 14, 10, 14, 18, 14
Arranging the data in ascending order such that same values are put together, we get:
10,12,12,14,14,14,14,14,14, 16, 16, 18.
Here, clearly, 14 occurs the most number of times.
Therefore, 14 is the mode of the given data.
Alternate solution:
Arranging the data in the form of a frequency table, we get:
Values  Tally Bars  Frequency 
10  ∣  1 
12  ∣∣  2 
14  6  
16  ∣∣  2 
18  ∣  1 
Total  12 
Clearly, 14 has maximum frequency. So, the mode of the given data is 14.
Heights of 25 children (in cm) in a school are as given below:
168, 165, 163, 160, 163, 161, 162, 164, 163, 162, 164, 163, 160, 163, 163, 165, 163, 162, 163, 164, 163, 160, 165, 163, 162
What is the mode of heights?
Also, find the mean and median.
Arranging the data in tabular form, we get:
Height of Children (cm)  Tally Bars  Frequency 
160  ∣∣∣  3 
161  ∣  1 
162  ∣∣∣∣  4 
163  10  
164  ∣∣∣  3 
165  ∣∣∣  3 
168  ∣  1 
Total  25 
Here, n = 25
∴ Median = Value of observation = Value of the 13^{th} observation = 163 cm.
Here, clearly, 163 cm occurs the most number of times. Therefore, the mode of the given data is 163 cm.
Now,
Mode = 3 Median  2 Mean
⇒ 163 = 3 x 163  2 Mean
⇒2 Mean = 326
⇒ Mean = 326 ÷ 2 = 163 cm.
The scores in mathematics test (out of 25) of 15 students are as follows:
19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20
Find the mode and median of this data. Are they same?
Arranging the data in ascending order such that same values are put together, we get:
5,9,10,12,15,16, 19, 20, 20, 20, 20, 23, 24, 25, 25.
Here, n = 15
∴ Median = Value of observation = Value of the 8^{th} observation = 20.
Here, clearly, 20 occurs the most number of times, i.e., 4 times. Therefore, the mode of the given data is 20.
Yes, the median and mode of the given data are the same.
Calculate the mean and median for the folllowing data:
Marks  :  10  11  12  13  14  16  19  20 
Number of students  :  3  5  4  5  2  3  2  1 
Using empirical formula, find its mode.
Calculation of Mean
Marks (x_{i})  10  11  12  13  14  16  19  20  Total 
Number of Students (f_{i})  3  5  4  5  2  3  2  1  
f_{i}x_{i}  30  55  48  65  28  48  38  20 
Mean =
Here, n = 25, which is an odd number. Therefore,
Median = Value of observation = the 13^{th} observation = 13.
Now,
Mode = 3 Median  2 Mean
⇒Mode = 3 x 13  2 x (13.28)
⇒Mode = 39  26.56
⇒Mode = 12.44.
The following table shows the weights of 12 persons.
Weight (in kg):  48  50  52  54  58 
Number of persons:  4  3  2  2  1 
Find the median and mean weights. Using empirical relation, calculate its mode.
Calculation of Mean
Weight (x_{i})  48  50  52  54  58  Total 
Number of Persons (f_{i})  4  3  2  2  1  
f_{i}x_{i}  192  150  104  108  58 
Here, n = 12
Now,
Mode = 3 Median  2 Mean
⇒ Mode = 3 x 50  2 x 51
⇒Mode = 150  102
⇒ Mode = 48 kg.
Thus, Mean = 51 kg, Median = 50 kg and Mode = 48 kg.
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