Statistics
Subject: Mathematics | Topic: Statistics
For TLT, ECT, PST, JEST & All Competitive Exams
Statistics is the branch of Mathematics that deals with collecting, organizing, presenting, analyzing and interpreting data. From exam results to population records and weather reports, statistics helps us make decisions from numbers. In competitive exams, questions mostly come from mean, median, mode, range, standard deviation, frequency tables and graphs.
1. Basic Terms
- Data: collected facts or figures (marks, heights, ages).
- Variable: a characteristic that can change from one person or thing to another.
- Raw (ungrouped) data: data in its original, unorganized form.
- Population: the whole group under study. Sample: a part selected from the population.
Types of Data
| Type | Meaning | Example |
|---|---|---|
| Qualitative | Describes quality, not numbers | Colour, gender, city |
| Quantitative (Discrete) | Counted, only whole values | Number of students |
| Quantitative (Continuous) | Measured, any value in a range | Height, weight, temperature |
| Primary data | Collected first-hand by the researcher | Own survey |
| Secondary data | Taken from existing sources | Census report |
2. Frequency Distribution
A frequency distribution is a table showing how often each value or class occurs. The number of times a value occurs is its frequency (f).
- Class interval: a group of values, e.g. 10–20.
- Class limits: the lowest and highest values of a class (10 and 20).
- Class width (h): upper limit − lower limit (20 − 10 = 10).
- Mid-point (x): (lower + upper) ÷ 2 = (10 + 20) ÷ 2 = 15.
- Class boundaries: the real limits that remove gaps between classes (e.g. 9.5–19.5).
- Cumulative frequency (cf): running total of frequencies.
- Relative frequency: f ÷ total frequency.
- Range: highest value − lowest value. Number of classes ≈ Range ÷ class width.
3. Measures of Central Tendency
These give one value that represents the centre of the data.
A. Arithmetic Mean (Average)
Frequency data: Mean = Σfx ÷ Σf (x = mid-point for grouped data)
Weighted mean = Σwx ÷ Σw
Properties: the sum of deviations from the mean is always zero; the mean is affected by extreme values; if every value is increased by k, the mean also increases by k; if every value is multiplied by k, the mean is multiplied by k.
B. Median (Middle Value)
Arrange data in order. If n is odd, median = the value at position (n + 1) ÷ 2. If n is even, median = average of the two middle values.
l = lower boundary of median class, cf = cumulative frequency before it, f = its frequency, h = class width.
The median is not affected by extreme values, so it is best for skewed data such as income.
C. Mode (Most Frequent Value)
The value that occurs most often. Data may have one mode (unimodal), two modes (bimodal), many modes, or no mode.
f₁ = frequency of modal class, f₀ = frequency before it, f₂ = frequency after it.
Empirical Relation
This holds approximately for moderately skewed data.
Other Means
- Geometric mean (G.M.) = ⁿ√(x₁ × x₂ × ... × xₙ). Used for growth rates.
- Harmonic mean (H.M.) = n ÷ Σ(1/x). Used for average speed.
- For positive numbers: A.M. ≥ G.M. ≥ H.M.
4. Solved Examples
Example 1 (ungrouped): Data: 2, 4, 4, 6, 9.
- Mean = 25 ÷ 5 = 5
- Median (middle of the ordered list) = 4
- Mode = 4
- Range = 9 − 2 = 7
Example 2 (grouped data):
| Class | f | Mid-point x | fx | cf |
|---|---|---|---|---|
| 0–10 | 5 | 5 | 25 | 5 |
| 10–20 | 8 | 15 | 120 | 13 |
| 20–30 | 12 | 25 | 300 | 25 |
| 30–40 | 5 | 35 | 175 | 30 |
| Total | 30 | 620 |
- Mean = 620 ÷ 30 = 20.67
- Median: n/2 = 15, so the median class is 20–30. Median = 20 + [(15 − 13) ÷ 12] × 10 = 21.67
- Mode: modal class is 20–30 (highest f = 12). Mode = 20 + [(12 − 8) ÷ (24 − 8 − 5)] × 10 = 20 + 40/11 = 23.64
Histogram of Example 2
The tallest bar (blue) is the modal class.
5. Measures of Dispersion (Spread)
Dispersion tells how far the data is spread around the centre.
| Measure | Formula |
|---|---|
| Range | Highest − Lowest |
| Coefficient of range | (H − L) ÷ (H + L) |
| Variance (σ²) | Σ(x − mean)² ÷ n |
| Standard deviation (σ) | √Variance |
| Coefficient of variation | (σ ÷ mean) × 100 |
Example 3: Data 2, 4, 4, 6, 9 with mean 5. Deviations: −3, −1, −1, 1, 4. Squares: 9, 1, 1, 1, 16, total 28. Variance = 28 ÷ 5 = 5.6. Standard deviation = √5.6 ≈ 2.37.
6. Quartiles
Quartiles divide ordered data into four equal parts: Q₁ (25%), Q₂ = median (50%), Q₃ (75%). Inter-quartile range = Q₃ − Q₁. Quartile deviation = (Q₃ − Q₁) ÷ 2.
7. Graphical Representation
| Graph | Used for |
|---|---|
| Bar chart | Comparing separate categories (bars have gaps) |
| Histogram | Continuous grouped data (bars touch, no gaps) |
| Frequency polygon | Joining mid-points of tops of histogram bars |
| Ogive (cumulative frequency curve) | Finding median and quartiles graphically |
| Pie chart | Showing parts of a whole |
| Line graph | Showing change over time |
Example: 40 students out of 200 → (40 ÷ 200) × 360 = 72°.
8. Quick Exam Tricks
- Always arrange data in order before finding the median.
- Mean of n numbers: total = mean × n. If one value is added or removed, use the total to recalculate.
- Median is best for skewed data; mean is best for symmetric data; mode is the only average for qualitative data.
- Histogram bars touch; bar chart bars do not.
- Sum of all pie-chart angles = 360°; sum of relative frequencies = 1.
Practice MCQs
1. The mean of 2, 4, 6, 8, 10 is:
a) 5 b) 6 c) 7 d) 8
2. The median of 3, 5, 7, 9 is:
a) 5 b) 7 c) 6 d) 8
3. The mode of 2, 3, 3, 4, 5, 3, 6 is:
a) 2 b) 3 c) 4 d) 5
4. Standard deviation is the ____ of variance.
a) square b) square root c) double d) half
5. A pie chart angle for 25% of the data is:
a) 45° b) 60° c) 90° d) 120°
Answers: 1-b, 2-c, 3-b, 4-b, 5-c
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