Statistics is the branch of mathematics dealing with the collection, presentation, analysis, and interpretation of data. In Class 10, we focus on finding the mean, median, and mode of grouped data, and drawing ogives (cumulative frequency curves).
---
Key Concepts
Frequency Distribution Table: Organises data into class intervals with corresponding frequencies.
Class Mark (Midpoint): xi = (Lower limit + Upper limit) / 2
---
Mean of Grouped Data
1. Direct Method:
Mean = (Sum of fi × xi) / (Sum of fi)
- 2. Assumed Mean Method:
- Choose assumed mean A (usually the midpoint of middle class)
- di = xi - A
- Mean = A + (Sum of fi di) / (Sum of fi)
- 3. Step Deviation Method:
- ui = (xi - A) / h (h = class width)
- Mean = A + h × (Sum of fi ui) / (Sum of fi)
---
Mode of Grouped Data
Modal Class: The class with the highest frequency.
Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] × h
- Where:
- l = lower limit of modal class
- f1 = frequency of modal class
- f0 = frequency of class preceding modal class
- f2 = frequency of class succeeding modal class
- h = class width
---
Median of Grouped Data
Median = l + [(n/2 - cf) / f] × h
- Where:
- l = lower limit of median class
- n = total frequency
- cf = cumulative frequency before median class
- f = frequency of median class
- h = class width
Median class: The class in which the cumulative frequency first exceeds n/2.
---
Empirical Relationship
Mode = 3 × Median - 2 × Mean
---
Worked Examples
Find the mean of the following data: 10-20 (f=5), 20-30 (f=8), 30-40 (f=15), 40-50 (f=12).
- Class marks: 15, 25, 35, 45
- f × x: 75, 200, 525, 540. Sum = 1340. Sum f = 40.
- Mean = 1340/40 = 33.5
Find the mode: 10-20 (2), 20-30 (5), 30-40 (8), 40-50 (3), 50-60 (2).
- Modal class = 30-40 (highest freq = 8), f1 = 8, f0 = 5, f2 = 3, l = 30, h = 10
- Mode = 30 + [(8-5)/(16-5-3)] × 10 = 30 + [3/8] × 10 = 30 + 3.75 = 33.75
Find the median if n = 40 and the cumulative frequencies reach 20 in class 30-40. If cf before = 17, f = 15, h = 10.
- Median = 30 + [(20-17)/15] × 10 = 30 + [3/15] × 10 = 30 + 2 = 32
If mean = 35 and median = 33, find mode using empirical formula.
- Mode = 3 × 33 - 2 × 35 = 99 - 70 = 29
Draw an ogive: An ogive is a graph of cumulative frequencies against upper class boundaries. It is used to find median, quartiles, and percentiles graphically.
- The median corresponds to the point on the ogive where cumulative frequency = n/2.
Find the mean using assumed mean method. Data: 0-10 (f=5), 10-20 (f=10), 20-30 (f=15), 30-40 (f=20), 40-50 (f=10). Assumed mean A = 25.
- d: -20, -10, 0, 10, 20. f×d: -100, -100, 0, 200, 200. Sum = 200. Sum f = 60.
- Mean = 25 + 200/60 = 25 + 3.33 = 28.33
The mean of the following data is 18. Find the missing frequency: 0-6 (6), 6-12 (8), 12-18 (10), 18-24 (x), 24-30 (6).
- Class marks: 3, 9, 15, 21, 27. Sum of f×x = 18 + 72 + 150 + 21x + 162 = 402 + 21x.
- Total frequency = 30 + x. Mean = (402 + 21x)/(30 + x) = 18.
- 402 + 21x = 540 + 18x => 3x = 138 => x = 46. Recalculate: 402 + 21(46) = 402 + 966 = 1368; 30 + 46 = 76; 1368/76 = 18. Correct!
---
Common mistakes
> Common mistakes: Students often forget to use class marks (not class limits) for mean calculations. For mode, ensure you identify the correct modal class before applying the formula. For median, first compute cumulative frequencies carefully, then identify the median class. Do not confuse n/2 with (n+1)/2 — for grouped data, use n/2.
---
Summary
Mean, median, and mode are the three measures of central tendency for grouped data. Use the direct method for mean when numbers are simple; use assumed mean or step deviation for larger data. The empirical formula Mode = 3 Median - 2 Mean is a useful shortcut. Ogives help find the median and quartiles graphically.