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compulsory · CSS study notes

8. Average

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 8 (Average)

Past Paper Weightage

  • Frequently tested as applied word problems (class scores, speed, mixed groups) — moderate-high frequency, quick to master.

Key Concepts

Average=Sum of all valuesNumber of values\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} Weighted Average=∑(value×weight)∑weight\text{Weighted Average} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weight}}
  • Combining averages of groups: total sum of group A + total sum of group B, divided by combined count (do NOT simply average the two averages unless group sizes are equal)
Average speed=Total distanceTotal time\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}

(NOT the simple average of two speeds, unless time spent at each speed is equal)

  • If one new value is added to a set:
New average=Old sum+New valueOld count+1\text{New average} = \frac{\text{Old sum} + \text{New value}}{\text{Old count} + 1}

Worked Example

  • Scores: 70, 80, 90 → average = 240/3 = 80
  • A car travels 60 km at 30 km/h and 60 km at 60 km/h → average speed = total distance/total time = 120/(2+1) = 40 km/h (NOT 45)

Past Paper Style Numericals (Solved)

  1. The average age of 30 students is 12 years. If the teacher's age (40 years) is included, find the new average.
    • Step 1: Find the total age of the 30 students: 30 × 12 = 360 years.
    • Step 2: Add the teacher's age: 360 + 40 = 400 years, and the new count of people is 30+1 = 31.
    • Step 3: New average = total ÷ count = 400 ÷ 31 ≈ 12.9 years.
  2. The average of 5 consecutive integers is 24. Find the largest integer.
    • Step 1: For consecutive integers, the average always equals the middle (3rd) value, so the middle number is 24.
    • Step 2: List the five numbers around the middle: 22, 23, 24, 25, 26.
    • Step 3: The largest integer is 26.
  3. A batsman's average after 10 innings is 42. After the 11th innings (scoring 64), find his new average.
    • Step 1: Find his total runs after 10 innings: 10 × 42 = 420 runs.
    • Step 2: Add the 11th innings score: 420 + 64 = 484 runs, over 11 innings.
    • Step 3: New average = 484 ÷ 11 = 44.
  4. Find the average of the first 20 natural numbers.
    • Step 1: Use the sum formula for the first n natural numbers: Sum = n(n+1)/2 = 20×21/2 = 210.
    • Step 2: Divide the sum by the count of numbers: 210 ÷ 20.
    • Step 3: Result: 10.5.

MCQ Traps

  • Average speed for equal distances ≠ average of the two speeds — a classic FPSC trap
  • Combining two groups' averages requires weighting by group size, not simple averaging

Self-Test Questions

  1. Five numbers average 20. If a sixth number of 32 is added, what is the new average?
  2. A cyclist rides 40 km at 20 km/h then 40 km at 40 km/h. Find the average speed for the whole trip.

Past-Paper MCQs (FPSC)

✅

The FPSC MCQ papers you shared (2024 and 2025) had no direct average question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2020 Q6(c): Ages of seven students: 17, 17, 18, 18, 18, 19, 19. Find the mean, median, mode and range, and define them. (5)

    1. Sum = 126, count = 7, so mean = 126 ÷ 7 = 18
    2. Sorted list, 4th value = median = 18
    3. Most frequent = mode = 18 (3 times)
    4. Range = 19 − 17 = 2
    • Definitions: mean = sum ÷ count; median = middle value of the sorted list; mode = most frequent value; range = largest − smallest

  • CSS 2016 Q10(b): Nine students aged 15, 15, 16, 16, 16, 17, 17, 18, 19. Calculate mean, median, mode and range. (5)

    1. Sum = 149, so mean = 149 ÷ 9 ≈ 16.56
    2. 5th value of 9 = median = 16
    3. Mode = 16 (appears 3 times)
    4. Range = 19 − 15 = 4

  • CSS 2021 Q7(b): The average of 11 numbers is 63; the first 6 average 60 and the last 6 average 65. Find the 6th number. (5)

    1. Total of all 11 = 11 × 63 = 693
    2. First six total = 360; last six total = 390
    3. The 6th number is counted in both groups, so 6th = (360 + 390) − 693
    4. 6th number = 57

  • CSS 2017 Q6(B)–(D): Hours teachers spend marking each day: 6 4 3 1 2 2 3 1 4 / 1 2 5 3 4 5 2 2 3 / 3 1 2 2 3 1 4 2. Make a frequency table and draw a histogram; how many teachers responded; longest and most common hours? (5 each)

    Hours123456
    Frequency586421
    • Histogram: draw 6 touching bars, heights 5, 8, 6, 4, 2, 1; label hours on the x-axis and number of teachers on the y-axis
    • Teachers who responded: 5 + 8 + 6 + 4 + 2 + 1 = 26
    • Longest number of hours: 6
    • Most common number of hours (mode): 2 hours (8 teachers)

  • CSS 2019 Q6(d): What do you understand by measures of central tendency? State its types. (5)

    • Measure of central tendency: a single value that represents the centre or typical value of a data set
    • Types
      • Arithmetic mean: sum of values ÷ number of values (affected by extreme values)
      • Median: middle value of the sorted data (average of the two middle values if the count is even); not affected by extremes
      • Mode: most frequent value; a set can have no mode, one, or several
      • Also: geometric mean (for growth rates) and harmonic mean (for rates such as speed)
    • Example: for 17, 17, 18, 18, 18, 19, 19 the mean, median and mode are all 18

  • CSS 2017 Q6(A): Differentiate with examples between a pictogram and a histogram. (5)

    FeaturePictogramHistogram
    ShowsData with pictures/symbols, each symbol standing for a stated number (a key)Frequency of numerical data grouped into class intervals
    Data typeCategories or simple countsContinuous (or grouped) numerical data
    Drawn withRows of iconsTouching bars (no gaps); bar width = class width, height = frequency
    ExampleCars sold each month, one car icon = 10 carsNumber of students in each marks range 0–10, 10–20…
    Best forQuick, simple, attractive displaySeeing the shape and spread of a distribution

  • CSS 2018 Q8(B): Define a histogram. How is it constructed? What information may be deduced from it? (5)

    • Histogram: a bar graph of a frequency distribution, with adjoining rectangles for each class interval
    • Construction
      1. Collect the data and find the range
      2. Choose class intervals of equal width
      3. Count how many values fall in each class (frequency table)
      4. Draw the x-axis (classes) and y-axis (frequency)
      5. Draw a bar for each class, with width = class width and height = frequency, with no gaps between bars
    • Information from it
      • Shape of the data (symmetric, skewed)
      • Modal class (tallest bar)
      • Range and spread; gaps and outliers
      • Total number of observations (sum of bar heights)
      • A rough estimate of the median and mean
    • Example: the 26 teacher responses in CSS 2017 Q6 give a histogram peaking at 2 hours

CSS Exam Perspective

🎯

  • Most repeated areas: mean, median, mode and range (2016, 2020), reverse average problems, frequency tables
  • Tip: sort the data first, and write the definitions, since they carry marks