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Q6:
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data:
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data:
Solution :
Given: A frequency distribution table representing the number of cars passing through a spot in 100 periods of 3 minutes each.
| Number of cars | Frequency ($f_i$) |
|---|---|
| 0 - 10 | 7 |
| 10 - 20 | 14 |
| 20 - 30 | 13 |
| 30 - 40 | 12 |
| 40 - 50 | 20 |
| 50 - 60 | 11 |
| 60 - 70 | 15 |
| 70 - 80 | 8 |
To find: The mode of the given grouped frequency distribution.
Step 1: Identify the Modal Class
The modal class is the class interval with the highest frequency. Observing the table, the highest frequency is $20$, which corresponds to the class interval $40 - 50$.
Therefore, the Modal Class is $40 - 50$.
Step 2: State the Formula for Mode
The formula for calculating the mode of grouped data is:
$\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
Where:
- $l$ = Lower limit of the modal class
- $h$ = Size of the class interval
- $f_1$ = Frequency of the modal class
- $f_0$ = Frequency of the class preceding the modal class
- $f_2$ = Frequency of the class succeeding the modal class
Step 3: Extract the Variables
Based on the modal class $40 - 50$:
- $l = 40$
- $h = 50 - 40 = 10$
- $f_1 = 20$
- $f_0 = 12$ (frequency of the class $30 - 40$)
- $f_2 = 11$ (frequency of the class $50 - 60$)
Step 4: Substitute and Calculate
Substituting the values into the formula:
$\text{Mode} = 40 + \left( \frac{20 - 12}{2(20) - 12 - 11} \right) \times 10$
$\text{Mode} = 40 + \left( \frac{8}{40 - 23} \right) \times 10$
$\text{Mode} = 40 + \left( \frac{8}{17} \right) \times 10$
$\text{Mode} = 40 + \frac{80}{17}$
Performing the division: $80 \div 17 \approx 4.7058...$
$\text{Mode} = 40 + 4.71$ (rounded to two decimal places)
$\text{Mode} = 44.71$
Final Answer: The mode of the data is 44.71 cars.
More Questions from Class 10 Mathematics Statistics EXERCISE 13.2
- Q1: The following table shows the ages of the patients admitted in a hospital during a year: Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
- Q2: The following data gives the information on the observed lifetimes (in hours) of 225 electrical components : Determine the modal lifetimes of the components.
- Q3: The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure :
- Q4: The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.
- Q5: The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches. Find the mode of the data.
CBSE Solutions for Class 10 Mathematics Statistics
Chapters in CBSE - Class 10 Mathematics
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