CODEBHAVYA โ€ข QUANTITATIVE APTITUDE

๐Ÿ“ˆ Averages

Understand arithmetic mean, weighted and combined averages, missing values and the fast methods used in aptitude tests, data analysis and programming.

๐ŸŽฏ 1. Learning Objectives

After completing this topic, you should be able to:

  • Calculate an arithmetic mean from a list of values.
  • Recover the total or a missing value from a known average.
  • Calculate weighted and combined averages correctly.
  • Handle addition, removal and replacement of observations.
  • Use shortcuts for consecutive and equally spaced numbers.
  • Distinguish ordinary average speed from equal-distance average speed.

๐Ÿ“– 2. Arithmetic Mean

Basic Meaning

An average represents an equal share of a total across all observations.

Average = Sum of observations / Number of observations

Total from Average

Rearranging the basic formula gives the total.

Total = Average ร— Number of observations
Example: Average of 10, 20 and 30 = (10+20+30)/3 = 20.

๐Ÿง  3. Useful Properties and Shortcuts

Same Change to Every Value

If every observation increases by k, the average also increases by k.

New average = Old average + k

Consecutive or Equally Spaced Values

For a symmetric sequence, the average is the middle value.

Average = (First + Last) / 2

Deviation Method

Choose an assumed mean A and average the deviations.

Mean = A + ฮฃd / n

Change in One Value

A total change ฮ” across n values changes the average by ฮ”/n.

Change in average = ฮ” / n

โš–๏ธ 4. Weighted Average

Use a weighted average when observations do not contribute equally.

Weighted average = ฮฃ(value ร— weight) / ฮฃweights
Example: score 80 with weight 2 and score 90 with weight 3 gives (80ร—2 + 90ร—3)/(2+3) = 86.

๐Ÿ‘ฅ 5. Combined Average

To combine groups, first convert each group average into its total.

Combined average = (nโ‚aโ‚ + nโ‚‚aโ‚‚) / (nโ‚ + nโ‚‚)
Example: 20 students average 60 and 30 students average 70. Combined average = (20ร—60 + 30ร—70)/50 = 66.

โ“ 6. Missing-Value Problems

  1. Find the required total: average ร— number of values.
  2. Add all known values.
  3. Subtract the known total from the required total.
Missing value = Required total โˆ’ Sum of known values

๐Ÿ”„ 7. Addition, Removal and Replacement

Adding a Value

Added value = New total โˆ’ Old total

Removing a Value

Removed value = Old total โˆ’ New total

Replacing a Value

New average = Old average + (Newโˆ’Old)/n

Wrong Entry Correction

Correct total = Recorded total โˆ’ Wrong + Correct

๐Ÿš— 8. Average Speed

General Rule

Average speed = Total distance / Total time

Equal Distances at x and y

Average speed = 2xy / (x+y)

Do not use (x+y)/2 unless the travel times are equal.

INTERACTIVE EXPLORER

๐Ÿงช 9. Average Explorer

Select a calculation, enter values and click Calculate Average. Input changes hide the previous result.

Select a calculation or example, then click Calculate Average.

๐Ÿงพ 10. Solved Examples

Example 1: Missing Number

Problem: Four numbers average 15. Three are 12, 18 and 14.

Solution: Required total=60; known total=44; missing=16.

Example 2: Added Observation

Problem: Five numbers average 24. Adding one makes the average 26.

Solution: Added value=6ร—26โˆ’5ร—24=36.

Example 3: Combined Group

Problem: 18 boys average 60. Twelve girls average 50.

Solution: Combined=(18ร—60+12ร—50)/30=56.

Example 4: Equal Distance Speed

Problem: Equal distances at 60 km/h and 40 km/h.

Solution: Average=2ร—60ร—40/(60+40)=48 km/h.

๐Ÿ’ป 11. Averages in Programming

Running Average

Programs update a mean as new sensor readings or records arrive.

Performance Measurement

Average response time, CPU usage and memory consumption summarize system behaviour.

Data Cleaning

Means help identify unusual values and can support missing-data estimates.

Machine Learning

Weighted averages combine losses, predictions and model outputs.

โš ๏ธ 12. Common Mistakes

Averaging Group Averages Directly

This works only when the groups have equal sizes.

Ignoring Weights

Values with different importance or frequency need a weighted mean.

Average Speed Error

For equal distances, ordinary arithmetic mean is usually wrong.

Forgetting Total = Average ร— Count

Most missing, addition and replacement problems begin with totals.

โœ๏ธ 13. Practice Problems

Solve each problem first, then use Show Solution to verify your method.

1. Find the average of 14, 18, 22 and 26.

2. The average of five values is 12. Find their total.

3. Four numbers average 25. Three are 20, 24 and 30. Find the fourth.

4. Ten employees average 35 years. One aged 40 is replaced by one aged 30.

5. Find the average speed for equal distances at 30 km/h and 60 km/h.

๐Ÿ“ 14. Quick Revision

  • Average = total รท number of observations.
  • Total = average ร— number of observations.
  • Weighted mean uses ฮฃ(valueร—weight)/ฮฃweights.
  • Combined mean converts each group average into a group total.
  • Missing value = required total โˆ’ known total.
  • A change ฮ” in one of n observations changes the average by ฮ”/n.
  • The mean of consecutive or equally spaced values is (first+last)/2.
  • Average speed always equals total distance รท total time.
INTERACTIVE PRACTICE

๐Ÿ“ˆ Averages Practice

Complete 20 questions from easy to hard. Your score, every option, correct answers and explanations appear only after final submission.

Start 20-Question Test โ†’