Methods to Find Mean ($\bar{x}$):
- Direct Method: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$
- Assumed Mean Method: $$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \quad \text{where } d_i = x_i - a$$
- Step-Deviation Method: $$\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h \quad \text{where } u_i = \frac{x_i - a}{h}$$
Find the mean marks obtained by students from the following distribution:
| Class Interval | Frequency ($f_i$) | Class Mark ($x_i$) | $f_i x_i$ |
|---|---|---|---|
| 10 - 25 | 2 | 17.5 | 35.0 |
| 25 - 40 | 3 | 32.5 | 97.5 |
| 40 - 55 | 7 | 47.5 | 332.5 |
| 55 - 70 | 6 | 62.5 | 375.0 |
| 70 - 85 | 6 | 77.5 | 465.0 |
| 85 - 100 | 6 | 92.5 | 555.0 |
| Total | $\sum f_i = 30$ | - | $\sum f_i x_i = 1860.0$ |
Solution:
Using the direct method formula:
$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$
$$\bar{x} = \frac{1860}{30} = 62$$
The mean mark is $62$.
Formula for Mode: $$\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$ 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
A survey conducted on 20 households in a locality resulted in the following frequency table for the number of family members:
Family size $1-3$ ($f=7$), $3-5$ ($f=8$), $5-7$ ($f=2$), $7-9$ ($f=2$), $9-11$ ($f=1$). Find the mode.
Solution:
Maximum frequency is $8$, belonging to class interval $3 - 5$.
Therefore, Modal Class = $3 - 5$.
Here: $l = 3$, $h = 2$, $f_1 = 8$, $f_0 = 7$, $f_2 = 2$.
$$\text{Mode} = 3 + \left( \frac{8 - 7}{2(8) - 7 - 2} \right) \times 2$$
$$\text{Mode} = 3 + \left( \frac{1}{16 - 9} \right) \times 2$$
$$\text{Mode} = 3 + \frac{2}{7} \approx 3 + 0.286 = 3.286$$
The mode of the given data is $3.286$.
Formula for Median: $$\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$$ where:
- $l$ = lower limit of median class
- $N$ = total number of observations ($\sum f_i$)
- $cf$ = cumulative frequency of class preceding the median class
- $f$ = frequency of the median class
- $h$ = class size