Statistics Solved Examples (10 Mathematics)

Statistics involves organizing, analyzing, and interpreting data using measures like mean, median, mode, and standard deviation. These examples cover group

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TL;DR: Statistics involves organizing, analyzing, and interpreting data using measures like mean, median, mode, and standard deviation. These examples cover…

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Statistics involves organizing, analyzing, and interpreting data using measures like mean, median, mode, and standard deviation. These examples cover group

Statistics — Solved Numerical Examples (Step by Step)

Example 1: Find the mean of the following data: 10, 15, 20, 25, 30, 35, 40

Solution: Mean = Sum of all observations / Number of observations

Sum = 10 + 15 + 20 + 25 + 30 + 35 + 40 = 175
Number of observations = 7

Mean = 175 / 7 = 25

Example 2: Find the median of: 12, 8, 25, 14, 30, 18, 22. Arrange in ascending order first.

Solution: Ascending order: 8, 12, 14, 18, 22, 25, 30

Number of observations = 7 (odd)
Median = Middle value = 4th observation = 18

(Formula for odd n: Median is at position (n+1)/2 = (7+1)/2 = 4th position)

Example 3: Find the mode of: 5, 7, 9, 5, 11, 7, 5, 13, 7, 15

Solution: Count the frequency of each value:
5 appears 3 times
7 appears 3 times
9 appears 1 time
11 appears 1 time
13 appears 1 time
15 appears 1 time

Both 5 and 7 have the highest frequency of 3.
This data is bimodal (has two modes).

Example 4: Calculate mean for grouped data: Class Interval: 0-10, 10-20, 20-30, 30-40 Frequency: 5, 8, 12, 6

Solution: Step 1: Find class mark (midpoint) for each interval.
Class Mark: 5, 15, 25, 35

Step 2: Multiply each class mark by its frequency.
5×5 = 25
15×8 = 120
25×12 = 300
35×6 = 210

Step 3: Find sum of (Class Mark × Frequency).
Sum = 25 + 120 + 300 + 210 = 655

Step 4: Sum of frequencies = 5 + 8 + 12 + 6 = 31

Mean = 655 / 31 = 21.13 (approximately)

Example 5: Find the range and variance of: 2, 4, 6, 8, 10

Solution: Step 1: Find range.
Range = Maximum - Minimum = 10 - 2 = 8

Step 2: Find mean.
Mean = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

Step 3: Find deviations from mean and square them.
(2-6)² = 16
(4-6)² = 4
(6-6)² = 0
(8-6)² = 4
(10-6)² = 16

Step 4: Find variance.
Variance = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8

Example 6: For the data 10, 20, 30, 40, 50, find the standard deviation.

Solution: Step 1: Find mean.
Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30

Step 2: Find squared deviations.
(10-30)² = 400
(20-30)² = 100
(30-30)² = 0
(40-30)² = 100
(50-30)² = 400

Step 3: Find variance.
Variance = (400 + 100 + 0 + 100 + 400) / 5 = 1000 / 5 = 200

Step 4: Find standard deviation.
Standard Deviation = √Variance = √200 = 10√2 ≈ 14.14

Example 7: In a class of 50 students, marks are grouped as: 0-20 (5 students), 20-40 (12 students), 40-60 (18 students), 60-80 (10 students), 80-100 (5 students). Find the median class and estimate median.

Solution: Step 1: Find cumulative frequencies.
0-20: 5
20-40: 5+12 = 17
40-60: 17+18 = 35
60-80: 35+10 = 45
80-100: 45+5 = 50

Step 2: Find median class.
Median position = n/2 = 50/2 = 25
The 25th observation falls in the 40-60 class (since CF = 35 > 25).
Median class = 40-60

Step 3: Use median formula.
Median = L + ((n/2 - CF) / f) × h
L = 40 (lower boundary of median class)
n/2 = 25
CF = 17 (cumulative frequency before median class)
f = 18 (frequency of median class)
h = 20 (class width)

Median = 40 + ((25 - 17) / 18) × 20
Median = 40 + (8/18) × 20
Median = 40 + 8.89 = 48.89

Tips

Always arrange data in ascending order before finding median and mode.
For grouped data, use class marks (midpoints) as representative values.
Mean is affected by extreme values (outliers), but median is more robust.
Standard deviation measures spread of data around the mean in the same units as the data.

Frequently Asked Questions

What is the difference between mean, median, and mode?

Mean is the average of all values. Median is the middle value when arranged in order. Mode is the value that appears most frequently. Use mean for symmetric data, median for skewed data or data with outliers, and mode for categorical data.

Why do we use variance and standard deviation?

Variance and standard deviation measure how spread out the data is from the mean. Standard deviation is preferred because it is in the same units as the original data, making it easier to interpret.

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