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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.
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.
Solution :
Given: The frequency distribution of ages of patients admitted to a hospital:
| Age (in years) | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65 |
|---|---|---|---|---|---|---|
| Number of patients ($f_i$) | 6 | 11 | 21 | 23 | 14 | 5 |
To Find: The Mode and the Mean of the given data, and to compare and interpret the results.
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 1.1: Identify the Modal Class. The maximum frequency is $23$, which corresponds to the class interval $35-45$. Thus, the modal class is $35-45$.
Step 1.2: Assign values. $l = 35$, $f_1 = 23$, $f_0 = 21$, $f_2 = 14$, $h = 10$.
Step 1.3: Substitute into the formula.
$\text{Mode} = 35 + \left( \frac{23 - 21}{2(23) - 21 - 14} \right) \times 10$
$\text{Mode} = 35 + \left( \frac{2}{46 - 35} \right) \times 10 = 35 + \left( \frac{2}{11} \right) \times 10 = 35 + \frac{20}{11} \approx 35 + 1.818 = 36.82$
Part 2: Calculation of Mean
Step 2.1: Prepare the table for Assumed Mean Method ($A = 40$).
| Age | $f_i$ | Class Mark ($x_i$) | $d_i = x_i - 40$ | $f_i d_i$ |
|---|---|---|---|---|
| 5-15 | 6 | 10 | -30 | -180 |
| 15-25 | 11 | 20 | -20 | -220 |
| 25-35 | 21 | 30 | -10 | -210 |
| 35-45 | 23 | 40 | 0 | 0 |
| 45-55 | 14 | 50 | 10 | 140 |
| 55-65 | 5 | 60 | 20 | 100 |
| Total | 80 | - | - | -370 |
Step 2.2: Apply the Mean formula.
$\text{Mean} (\bar{x}) = A + \frac{\sum f_i d_i}{\sum f_i} = 40 + \left( \frac{-370}{80} \right) = 40 - 4.625 = 35.375$
Part 3: Comparison and Interpretation
The modal age is approximately $36.82$ years, representing the age group with the highest number of patients admitted. The mean age is $35.38$ years, representing the average age of all patients admitted. Since the mean is slightly less than the mode, it indicates that the distribution is slightly negatively skewed, but both measures suggest that the majority of patients admitted are in the mid-30s age range.
Final Answer: The Mode is 36.82 years and the Mean is 35.38 years.
More Questions from Class 10 Mathematics Statistics EXERCISE 13.2
- 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.
- 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:
CBSE Solutions for Class 10 Mathematics Statistics
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