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Mean and Deviation Basics

Introduction

Mean and deviation form the foundation of understanding how data values cluster and spread. The mean (average) gives a central value, while deviations show how far each observation lies from that centre. This pattern is important because it helps you measure consistency and spot outliers quickly.

Pattern: Mean and Deviation Basics

Pattern

The key concept: The mean gives the central tendency; deviation = (observation - mean) shows each value's distance from that centre.

Step-by-Step Example

Question

Find the mean of the numbers 8, 12, and 15, and then calculate the deviation of each number from the mean.

Solution

  1. Step 1: Identify the given data

    Observations: 8, 12, 15
    Number of observations (n) = 3

  2. Step 2: Apply the mean formula

    Mean (x̄) = (Sum of observations) ÷ n
    = (8 + 12 + 15) ÷ 3
    = 35 ÷ 3 = 11.67

  3. Step 3: Compute deviations for each observation

    Deviation of 8 = 8 - 11.67 = -3.67
    Deviation of 12 = 12 - 11.67 = +0.33
    Deviation of 15 = 15 - 11.67 = +3.33

  4. Final Answer:

    Mean = 11.67
    Deviations: 8 → -3.67, 12 → +0.33, 15 → +3.33

  5. Quick Check:

    Sum of deviations = (-3.67 + 0.33 + 3.33) = 0 ✅ Deviations always sum to zero - calculation verified.

Quick Variations

1. Apply the same logic to any number of ungrouped data points.

2. For grouped data, use midpoints (class marks) to find the mean and deviations.

3. When data values are large or repetitive, use an assumed mean (A) to simplify calculations.

Trick to Always Use

  • Step 1: Quickly find the mean by dividing the total sum by the number of observations.
  • Step 2: Subtract the mean from each observation to find deviations.
  • Step 3: Verify your result - the sum of deviations must equal zero (a simple accuracy check).

Summary

Summary

In the Mean and Deviation Basics pattern:

  • The mean represents the central or average value of a dataset.
  • The deviation of each observation = (observation - mean) shows how far each value lies from the mean.
  • The sum of all deviations from the mean is always zero.
  • This property helps verify your calculations and forms the base for variance and standard deviation.
  • Always double-check that your deviations add up to zero - it’s the quickest correctness test.

Practice

(1/5)
1. Find the mean of the numbers 10, 20, and 30.
easy
A. 20
B. 25
C. 15
D. 30

Solution

  1. Step 1: Identify the given data

    Identify the given data: 10, 20, 30.

  2. Step 2: Apply the mean formula

    Mean = (10 + 20 + 30) ÷ 3 = 60 ÷ 3 = Mean = 20 → Option A.

  3. Final Answer:

    Mean = 20 → Option A.

  4. Quick Check:

    (10 + 20 + 30) ÷ 3 = 20 ✅
Hint: Add all numbers and divide by total count.
Common Mistakes: Dividing by 2 instead of 3 or forgetting one number.
2. The mean of 5, 10, and 15 is?
easy
A. 8
B. 10
C. 12
D. 15

Solution

  1. Step 1: List the data values

    List data: 5, 10, 15.

  2. Step 2: Calculate the mean

    Mean = (5 + 10 + 15) ÷ 3 = 30 ÷ 3 = Mean = 10 → Option B.

  3. Final Answer:

    Mean = 10 → Option B.

  4. Quick Check:

    (5 + 10 + 15) ÷ 3 = 10 ✅
Hint: Add numbers, divide by count.
Common Mistakes: Taking median (10) as mean without checking formula.
3. For the numbers 4, 6, 8, find the deviations from their mean.
easy
A. −1, 0, +1
B. −3, 0, +3
C. −2, 0, +2
D. −4, 0, +4

Solution

  1. Step 1: Find the mean

    Find mean = (4 + 6 + 8) ÷ 3 = 18 ÷ 3 = 6.

  2. Step 2: Compute deviations from the mean

    Compute deviations: 4 - 6 = -2; 6 - 6 = 0; 8 - 6 = +2.

  3. Final Answer:

    Deviations = -2, 0, +2 → Option C.

  4. Quick Check:

    Sum of deviations = (-2 + 0 + 2) = 0 ✅
Hint: Subtract mean from each number; deviations should balance around zero.
Common Mistakes: Subtracting in reverse (mean - value) and getting opposite signs.
4. The mean of 3, 7, 11, and 15 is 9. Find the deviation of each value from the mean.
medium
A. −4, −2, +2, +4
B. −3, −1, +1, +3
C. −6, −4, +4, +6
D. −6, −2, +2, +6

Solution

  1. Step 1: Note the given mean

    Given mean = 9.

  2. Step 2: Subtract mean from each value

    Compute deviations: 3 - 9 = -6; 7 - 9 = -2; 11 - 9 = +2; 15 - 9 = +6.

  3. Final Answer:

    Deviations = -6, -2, +2, +6 → Option D.

  4. Quick Check:

    Sum = (-6 - 2 + 2 + 6) = 0 ✅
Hint: Subtract mean from each number; confirm that sum of deviations = 0.
Common Mistakes: Using wrong mean or forgetting negative signs.
5. For the numbers 6, 8, 10, 12, and 14, find the mean and check if the deviations sum to zero.
medium
A. Mean = 10; Sum = 0
B. Mean = 9; Sum = 1
C. Mean = 10; Sum = 2
D. Mean = 8; Sum = 0

Solution

  1. Step 1: Calculate the mean

    Find mean = (6 + 8 + 10 + 12 + 14) ÷ 5 = 50 ÷ 5 = 10.

  2. Step 2: Compute deviations

    Find deviations: (6 - 10 = -4), (8 - 10 = -2), (10 - 10 = 0), (12 - 10 = +2), (14 - 10 = +4).

  3. Step 3: Sum the deviations

    Sum of deviations = (-4 - 2 + 0 + 2 + 4) = 0.

  4. Final Answer:

    Mean = 10; Sum of deviations = 0 → Option A.

  5. Quick Check:

    Sum of deviations = 0 confirms mean is correct ✅
Hint: Evenly spaced data → mean = middle value, deviations cancel out.
Common Mistakes: Adding deviations without checking sign properly.

Mock Test

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