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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.
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.
Solution :
Given: The frequency distribution of the number of students per teacher in various states/UTs of India:
| Number of students per teacher | Number of states/U.T. ($f_i$) |
|---|---|
| 15 - 20 | 3 |
| 20 - 25 | 8 |
| 25 - 30 | 9 |
| 30 - 35 | 10 |
| 35 - 40 | 3 |
| 40 - 45 | 0 |
| 45 - 50 | 0 |
| 50 - 55 | 2 |
To Find: The Mode and the Mean of the given data and provide an interpretation.
Part 1: Calculation of Mode
Formula: $\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
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 1: Identify the Modal Class. The highest frequency is $10$, which corresponds to the class interval $30 - 35$. Thus, the modal class is $30 - 35$.
Step 2: Assign values. $l = 30$, $f_1 = 10$, $f_0 = 9$, $f_2 = 3$, $h = 5$.
Step 3: Substitute into the formula.
$\text{Mode} = 30 + \left( \frac{10 - 9}{2(10) - 9 - 3} \right) \times 5$
$\text{Mode} = 30 + \left( \frac{1}{20 - 12} \right) \times 5 = 30 + \frac{5}{8} = 30 + 0.625 = 30.625$
Part 2: Calculation of Mean
We use the Assumed Mean Method: $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$, where $d_i = x_i - a$. Let assumed mean $a = 32.5$.
| Class Interval | $f_i$ | Class mark ($x_i$) | $d_i = x_i - 32.5$ | $f_i d_i$ |
|---|---|---|---|---|
| 15-20 | 3 | 17.5 | -15 | -45 |
| 20-25 | 8 | 22.5 | -10 | -80 |
| 25-30 | 9 | 27.5 | -5 | -45 |
| 30-35 | 10 | 32.5 | 0 | 0 |
| 35-40 | 3 | 37.5 | 5 | 15 |
| 40-45 | 0 | 42.5 | 10 | 0 |
| 45-50 | 0 | 47.5 | 15 | 0 |
| 50-55 | 2 | 52.5 | 20 | 40 |
| Total | 35 | - | - | -115 |
Step 4: Calculate Mean.
$\bar{x} = 32.5 + \left( \frac{-115}{35} \right) = 32.5 - 3.2857 \approx 29.21$
Interpretation
The mode of $30.6$ indicates that the most common teacher-student ratio in the majority of states is approximately $30.6$. The mean of $29.2$ indicates that, on average, there are about $29$ students per teacher across the states.
Final Answer: Mode = 30.625, Mean = 29.21
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 :
- 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.
- 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:
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