Statistics : Exercise of mean median and mode

📊 MEAN MEDIAN MODE 📈
Grade 8 Statistics Guide

Mean, Median, and Mode - Complete Guide for Grade 8

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📊 MEAN

Definition

The mean (also called average) is the sum of all values divided by the number of values in a data set.

Formula for Individual Series

$\stackrel{\mathbf{-}}{\mathit{X}}\mathbf{ }\mathbf{=}\frac{\mathit{\Sigma }\mathit{x}}{\mathit{N}}$

Where:
• Σx = Sum of all observations
• n = Number of observations
• x̄ = Mean (read as "x-bar")

Practice Questions - Finding Mean

Question 1: Find the mean of: 5, 8, 12, 15, 20

Question 2: Calculate the mean of the following marks: 78, 85, 92, 67, 88, 74

Question 3: The heights (in cm) of 7 students are: 150, 155, 148, 162, 158, 145, 152. Find the mean height.

Question 4: Find the mean of: 3.5, 4.2, 6.8, 5.1, 7.3, 4.9

Question 5: The scores in a basketball game were: 24, 18, 32, 28, 16, 22, 30, 26. Calculate the mean score.

Question 6: Find the mean of the first 8 natural numbers.

Practice Questions - Finding Unknown Value (x)

Question 7: The mean of 6, 8, 10, 12, x is 10. Find the value of x.

Question 8: If the mean of 15, 20, 25, 30, x, 35 is 25, find x.

Question 9: The mean of five numbers is 18. Four of the numbers are 12, 16, 20, and 24. Find the fifth number.

Question 10: The average age of 8 students is 14 years. If 7 students have ages 12, 13, 15, 14, 16, 13, 15, find the age of the 8th student.

Question 11: The mean of 7, 9, x, 13, 15 is 11. Calculate the value of x.

Answers - Mean

Answer 1:
Mean = (5 + 8 + 12 + 15 + 20) ÷ 5 = 60 ÷ 5 = 12

Answer 2:
Mean = (78 + 85 + 92 + 67 + 88 + 74) ÷ 6 = 484 ÷ 6 = 80.67

Answer 3:
Mean = (150 + 155 + 148 + 162 + 158 + 145 + 152) ÷ 7 = 1070 ÷ 7 = 152.86 cm

Answer 4:
Mean = (3.5 + 4.2 + 6.8 + 5.1 + 7.3 + 4.9) ÷ 6 = 31.8 ÷ 6 = 5.3

Answer 5:
Mean = (24 + 18 + 32 + 28 + 16 + 22 + 30 + 26) ÷ 8 = 196 ÷ 8 = 24.5

Answer 6:
First 8 natural numbers: 1, 2, 3, 4, 5, 6, 7, 8
Mean = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) ÷ 8 = 36 ÷ 8 = 4.5

Answer 7:
Mean = (6 + 8 + 10 + 12 + x) ÷ 5 = 10
(36 + x) ÷ 5 = 10
36 + x = 50
x = 14

Answer 8:
Mean = (15 + 20 + 25 + 30 + x + 35) ÷ 6 = 25
(125 + x) ÷ 6 = 25
125 + x = 150
x = 25

Answer 9:
Sum of five numbers = 18 × 5 = 90
Sum of four known numbers = 12 + 16 + 20 + 24 = 72
Fifth number = 90 - 72 = 18

Answer 10:
Total age of 8 students = 14 × 8 = 112 years
Sum of 7 known ages = 12 + 13 + 15 + 14 + 16 + 13 + 15 = 98 years
Age of 8th student = 112 - 98 = 14 years

Answer 11:
Mean = (7 + 9 + x + 13 + 15) ÷ 5 = 11
(44 + x) ÷ 5 = 11
44 + x = 55
x = 11

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📐 MEDIAN

Definition

The median is the middle value when data is arranged in ascending or descending order. If there are two middle values, the median is their average.

Formula for Individual Series

$$\begin{gathered} \mathit{M}\mathit{e}\mathit{d}\mathit{i}\mathit{a}\mathit{n}\mathbf{ }\mathbf{=}\mathit{M}\mathit{i}\mathit{d}\mathit{d}\mathit{l}\mathit{e} \mathit{t}\mathit{e}\mathit{r}\mathit{m} \\ \mathit{o}\mathit{r} {\left(\frac{\mathbf{\text{N + 1}}}{\mathbf{\text{2}}}\right)}^{\mathbf{\text{th}}}\mathbf{\text{items}} \end{gathered}$$

Practice Questions - Finding Median

Question 1: Find the median of: 7, 3, 9, 5, 11, 2, 8

Question 2: Calculate the median of: 45, 67, 23, 78, 56, 34

Question 3: Find the median of the marks: 85, 92, 78, 88, 95, 82, 90, 87, 89

Question 4: The weights (in kg) of 6 students are: 45, 52, 48, 55, 49, 51. Find the median weight.

Question 5: Find the median of: 12, 18, 25, 19, 22, 16, 28, 21

Question 6: Calculate the median of the first 9 prime numbers.

Practice Questions - Finding Unknown Value (x)

Question 7: The median of 8, 12, x, 16, 20 (in ascending order) is 12. Find x.

Question 8: If the median of 5, 7, x, 13, 15, 18 is 10, find the value of x.

Question 9: The median of 3, 6, 9, x, 15, 18 is 10. Find x.

Question 10: Find x if the median of 4, 6, 8, x, 12, 14 is 9.

Question 11: The median of 2, 5, x, 8, 11 is 7. Calculate x.

Answers - Median

Answer 1:
Arranged in order: 2, 3, 5, 7, 8, 9, 11
n = 7 (odd), Position = (7+1)/2 = 4th
Median = 7

Answer 2:
Arranged in order: 23, 34, 45, 56, 67, 78
n = 6 (even), Positions = 3rd and 4th values
Median = (45 + 56)/2 = 50.5

Answer 3:
Arranged in order: 78, 82, 85, 87, 88, 89, 90, 92, 95
n = 9 (odd), Position = (9+1)/2 = 5th
Median = 88

Answer 4:
Arranged in order: 45, 48, 49, 51, 52, 55
n = 6 (even), Positions = 3rd and 4th values
Median = (49 + 51)/2 = 50 kg

Answer 5:
Arranged in order: 12, 16, 18, 19, 21, 22, 25, 28
n = 8 (even), Positions = 4th and 5th values
Median = (19 + 21)/2 = 20

Answer 6:
First 9 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23
n = 9 (odd), Position = (9+1)/2 = 5th
Median = 11

Answer 7:
Data in order: 8, 12, x, 16, 20
For median to be 12, x must be ≤ 12
Since median = 12 (middle value), x = 12

Answer 8:
n = 6 (even), median = average of 3rd and 4th values = 10
Data in order: 5, 7, x, 13, 15, 18
For median = 10: (x + 13)/2 = 10
x + 13 = 20
x = 7

Answer 9:
Data in order: 3, 6, 9, x, 15, 18
n = 6 (even), median = (9 + x)/2 = 10
9 + x = 20
x = 11

Answer 10:
Data in order: 4, 6, 8, x, 12, 14
n = 6 (even), median = (8 + x)/2 = 9
8 + x = 18
x = 10

Answer 11:
Data in order: 2, 5, x, 8, 11
n = 5 (odd), median = x = 7
Therefore, x = 7

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🎯 MODE

Definition

The mode is the value that appears most frequently in a data set. A data set can have no mode, one mode, or multiple modes.

Types

• No Mode: All values appear with the same frequency
• Unimodal: One value appears most frequently
• Bimodal: Two values appear with the highest frequency
• Multimodal: More than two values appear with the highest frequency

Practice Questions - Finding Mode

Question 1: Find the mode of: 3, 5, 7, 3, 9, 3, 2, 5

Question 2: Calculate the mode of: 12, 15, 18, 12, 20, 15, 12, 18

Question 3: Find the mode of the scores: 85, 92, 78, 85, 90, 78, 85, 88

Question 4: What is the mode of: 4, 6, 8, 10, 12, 14, 16?

Question 5: Find the mode of: 25, 30, 25, 35, 30, 40, 25, 30

Question 6: Calculate the mode of: 2.5, 3.8, 2.5, 4.1, 3.8, 5.2, 2.5

Practice Questions - Finding Unknown Value (x)

Question 7: In the data 8, 12, x, 16, 8, 20, the mode is 8. What are the possible values of x?

Question 8: The mode of 5, 7, 9, x, 7, 11 is 7. Find all possible values of x.

Question 9: If x is added to the data 3, 5, 7, 5, 9, and the mode becomes bimodal, find the possible values of x.

Question 10: In the data set 6, 8, 10, x, 8, 12, if the mode is 8, what values can x take?

Question 11: The data 4, 6, x, 8, 6, 10 has mode 6. Find the range of values for x.

Answers - Mode

Answer 1:
Count: 3 appears 3 times, 5 appears 2 times, others appear once
Mode = 3

Answer 2:
Count: 12 appears 3 times, 15 appears 2 times, 18 appears 2 times
Mode = 12

Answer 3:
Count: 85 appears 3 times, 78 appears 2 times, others appear once
Mode = 85

Answer 4:
All values appear once
No mode

Answer 5:
Count: 25 appears 3 times, 30 appears 3 times
Bimodal: 25 and 30

Answer 6:
Count: 2.5 appears 3 times, 3.8 appears 2 times, others appear once
Mode = 2.5

Answer 7:
For mode to be 8, x can be any value except 12, 16, or 20 (which would create another mode)
Possible values: x ≠ 12, 16, 20

Answer 8:
For mode to remain 7, x can be any value except 5, 9, or 11
Possible values: x ≠ 5, 9, 11

Answer 9:
Current data: 3, 5, 7, 5, 9 (mode = 5)
For bimodal: x = 3, 7, or 9 (to make frequency = 2 each)

Answer 10:
For mode to be 8, x can be any value except 6, 10, or 12
Possible values: x ≠ 6, 10, 12

Answer 11:
For mode to remain 6, x can be any value except 4, 8, or 10
Range: x ≠ 4, 8, 10

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🧠 QUIZ TIME!

Question 1: The mean of 24, 28, x, 32, 36 is 30. Find x.

a) 30 b) 28 c) 32 d) 26

Question 2: Find the median of: 15, 22, 18, 25, 20, 19, 21

a) 20 b) 19 c) 21 d) 18

Question 3: What is the mode of: 6, 8, 6, 10, 8, 12, 6, 8?

a) 6 only b) 8 only c) Both 6 and 8 d) No mode

Question 4: If the median of 5, 8, x, 12, 15 is 10, then x =?

a) 10 b) 8 c) 12 d) 9

Question 5: The mean of first 5 odd numbers is: