Class 10 Maths Notes Chapter 13 — Statistics#
Key terms#
- Class mark $x_i$ — the mid-point of a class: $$x_i = \frac{\text{upper limit} + \text{lower limit}}{2}$$
- Class size $h$ — upper limit minus lower limit.
- Cumulative frequency ($cf$) of a class — the sum of the frequencies of that class and all classes before it.
- Modal class — the class with the highest frequency.
- Median class — the first class whose cumulative frequency is greater than $\frac{n}{2}$.
Mean of grouped data#
Each class is represented by its class mark. Use any one method:
Direct method:
$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$Assumed mean method, with $d_i = x_i - a$:
$$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$$Step deviation method, with $u_i = \frac{x_i - a}{h}$:
$$\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h$$
- All three give the same mean.
- Take the assumed mean $a$ as a class mark near the middle of the table.
- Use the direct method when $x_i$ and $f_i$ are small. Use step deviation when the numbers are large and the classes are of equal size.
Mode of grouped data#
$$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$
- $l$ = lower limit of the modal class, $h$ = class size
- $f_1$ = frequency of the modal class
- $f_0$ = frequency of the class before it, $f_2$ = frequency of the class after it
Median of grouped data#
$$\text{Median} = l + \frac{\frac{n}{2} - cf}{f} \times h$$
- $n$ = total frequency, $l$ = lower limit of the median class
- $cf$ = cumulative frequency of the class before the median class
- $f$ = frequency of the median class, $h$ = class size
Before using the mode or median formula, make sure the classes are continuous. For classes like 118–126, 127–135, use the class boundaries 117.5–126.5, 126.5–135.5, and so on.
Empirical relation#
$$3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$$This relation is approximate and lets you find one measure from the other two.
Which measure to use#
| Measure | Best when |
|---|---|
| Mean | All observations matter and there are no extreme values (average marks) |
| Median | There are extreme values and a “typical” value is wanted (average wage) |
| Mode | The most frequent or most popular item is wanted (most-watched TV show) |
Common mistakes#
- Using the class limits instead of the class mark for $x_i$.
- In the mode formula, using the cumulative frequencies instead of the class frequencies.
- In the median formula, using the cumulative frequency of the median class instead of the class before it.
- Using $f_0$ and $f_2$ from the wrong classes. They are the classes immediately before and after the modal class.
- For “find the missing frequency”, writing $\sum f_i$ without the unknown. Include it, like $44 + f$.