Class 10 NCERT Maths Chapter 13: Statistics
In this chapter, we study how to collect, organise, and interpret data, and how to represent it using measures such as Mean, Median, and Mode.
1. Mean of Grouped Data
Formula (Direct Method):
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] where \(f_i\) = frequency, \(x_i\) = class mark = \(\frac{\text{upper limit} + \text{lower limit}}{2}\)
Assumed Mean Method:
\[ \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \] where \(d_i = x_i - a\), and \(a\) is the assumed mean.
Step Deviation Method:
\[ \bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h \] where \(u_i = \frac{x_i - a}{h}\) and \(h\) = class width.
2. Median of Grouped Data
Formula:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \]
- \(L\) = lower boundary of median class
- \(N\) = total frequency = \(\sum f_i\)
- \(CF\) = cumulative frequency before median class
- \(f\) = frequency of median class
- \(h\) = class width
3. Mode of Grouped Data
Formula:
\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]
- \(L\) = lower boundary of modal class
- \(f_1\) = frequency of modal class
- \(f_0\) = frequency of class before modal class
- \(f_2\) = frequency of class after modal class
- \(h\) = class width
4. Example Table (for practice)
| Class Interval | Frequency (\(f_i\)) | Class Mark (\(x_i\)) | \(f_i x_i\) |
|---|---|---|---|
| 0 – 10 | 5 | 5 | 25 |
| 10 – 20 | 8 | 15 | 120 |
| 20 – 30 | 10 | 25 | 250 |
| 30 – 40 | 7 | 35 | 245 |
| 40 – 50 | 5 | 45 | 225 |
\[ \sum f_i = 35,\quad \sum f_i x_i = 865 \] So, \[ \bar{x} = \frac{865}{35} \approx 24.71 \]
5. Important Tips
- Always arrange data in ascending order of class intervals.
- Choose an assumed mean that simplifies calculations.
- Median and mode require identifying the correct class first.
- Write all steps neatly in exams for full marks.