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CBSE · Class 11 · Mathematics

Statistics: Range and Mean Deviation (Ch 13)

4 concepts · 32 practice questions with answers · Free to read

Dispersion and range

From daily life

Two cricketers both average 45 runs. One scores between 40 and 50 every match; the other scores anything from 10 to 80. The average is the same, but the batting is very different. Dispersion measures this spread.

Simple definition

Dispersion tells how far the data is scattered around a central value. The simplest measure is the range = largest value − smallest value. Two data sets can have the same mean and very different ranges.

Formula: Range = Largest − Smallest

Rule 1: Find the range — Pick the largest and the smallest value.

  • Easy: Data 40, 42, 45, 48, 50: range = 50 − 40 = 10.
  • Medium: Data 10, 30, 45, 60, 80: range = 80 − 10 = 70, though the mean is 45 for both.
  • Exam-type: Data 7.5, 12, 3.5, 9: range = 12 − 3.5 = 8.5.

Rule 2: Use of range — A big range means a widely spread set.

  • Easy: Marks of class X: range 10 (marks are close together).
  • Medium: Marks of class Y: range 60 (marks are widely spread).
  • Exam-type: Range uses only two values, so one odd value changes it a lot.
Remember

Range is quick but ignores all values except the two extremes.

Common mistake

Subtracting the first and last values as they are written, without sorting out the smallest and largest.

Memory clue

Range = Big minus Small.

Practice questions

Easy
  1. 1Find the range of 8, 15, 3, 20, 11.
  2. 2Find the range of 45, 52, 38, 60.
  3. 3Class A marks 55, 58, 60, 62, 65 and class B marks 30, 45, 60, 75, 90. Which has the larger range?
Medium
  1. 4The highest mark is 92 and the range is 35. Find the lowest mark.
  2. 5Set P: 20, 22, 24, 26, 28 and set Q: 5, 15, 24, 33, 43. Both have mean 24. Compare the ranges.
  3. 6Find the range of 12.5, 7.5, 20, 9.
Challenging
  1. 7Every value of a data set is increased by 6 and the old range was 20. Find the new range.
  2. 8Every value of a data set is multiplied by 3 and the old range was 14. Find the new range.
Show answers (try them yourself first)
  1. 117
  2. 222
  3. 3B (60 against 10)
  4. 457
  5. 58 and 38, so Q is more spread
  6. 612.5
  7. 720 (unchanged)
  8. 842

Mean deviation about the mean

From daily life

A bus driver wants to know how far, on average, each trip time differs from the usual time. Add up the gaps (ignoring plus or minus) and average them.

Simple definition

The mean deviation about the mean is the average of the absolute gaps between each value and the mean. Absolute gap means we ignore the sign.

Formula: MD = (sum of |x − mean|) ÷ n

Rule 1: Steps — Mean, then gaps, then absolute values, then their average.

  • Easy: 6, 9, 12, 15, 18: mean 12; gaps 6.0, 3.0, 0.0, 3.0, 6.0; sum 18; MD = 3.6.
  • Medium: 5, 8, 8, 11, 13, 15: mean 10; gaps 5.0, 2.0, 2.0, 1.0, 3.0, 5.0; sum 18; MD = 3.
  • Exam-type: 10, 14, 18, 22, 26, 30, 34: mean 22; gaps 12.0, 8.0, 4.0, 0.0, 4.0, 8.0, 12.0; sum 48; MD = 48/7 ≈ 6.86.

Rule 2: Why absolute — Plain gaps add up to 0, so we use absolute gaps.

  • Easy: The signed gaps of 6, 9, 12, 15, 18 are −6, −3, 0, 3, 6, which add to 0.
  • Medium: A larger MD means a more scattered set.
  • Exam-type: Adding the same number to every value does not change the MD.
Remember

The sum of signed deviations from the mean is always 0.

Common mistake

Forgetting the absolute sign and getting 0.

Memory clue

Gap, ignore sign, average.

Practice questions

Easy
  1. 1Find the mean deviation about the mean of 4, 6, 8, 10, 12.
  2. 2Find the mean deviation about the mean of 3, 7, 5, 9, 6.
  3. 3Find the mean deviation about the mean of 20, 25, 30.
Medium
  1. 4Find the mean deviation about the mean of 2, 5, 5, 8, 10, 14.
  2. 5Find the mean deviation about the mean of 11, 13, 17, 19, 25, 27.
  3. 6Find the mean deviation about the mean of the first five natural numbers.
Challenging
  1. 7The sum of absolute deviations of 8 numbers from their mean is 48. Find the mean deviation.
  2. 8For 10 values the mean is 25 and the MD is 4. Each value is increased by 5. Find the new mean and MD.
Show answers (try them yourself first)
  1. 12.4
  2. 21.6
  3. 33.33
  4. 43.33
  5. 55
  6. 61.2
  7. 76
  8. 8Mean 30, MD 4

Mean deviation about the median

From daily life

When one value is very large, the mean is pulled towards it. The median is steadier, so people also measure the gaps from the median.

Simple definition

Arrange the data in order and find the median (middle value, or the average of the two middle values). Then MD about median = (sum of |x − median|) ÷ n. It is the smallest possible mean deviation.

Formula: MD (median) = (sum of |x − median|) ÷ n

Rule 1: Odd number of values — Median is the middle value.

  • Easy: 3, 5, 9, 11, 17: median 9; gaps 6, 4, 0, 2, 8; sum 20; MD = 4.
  • Medium: 8, 12, 15, 20, 25: median 15; gaps 7, 3, 0, 5, 10; sum 25; MD = 5.
  • Exam-type: Median first, always after arranging in order.

Rule 2: Even number of values — Median is the average of the two middle values.

  • Easy: 4, 6, 10, 12: median 8; gaps 4, 2, 2, 4; MD = 3.
  • Medium: 2, 4, 8, 10, 16, 20: median 9; gaps 7, 5, 1, 1, 7, 11; sum 32; MD = 5.33.
  • Exam-type: 1, 2, 3, 4, 10: MD about mean = 2.4, about median = 2.2; the median gives the smaller value.
Remember

Arrange the data in increasing order before taking the median.

Common mistake

Taking the middle value of the data in the order it was written.

Memory clue

Order first, middle next, gaps last.

Practice questions

Easy
  1. 1Find the mean deviation about the median of 3, 5, 9, 11, 17.
  2. 2Find the mean deviation about the median of 8, 12, 15, 20, 25.
  3. 3Find the mean deviation about the median of 4, 6, 10, 12.
Medium
  1. 4Find the mean deviation about the median of 2, 4, 8, 10, 16, 20.
  2. 5Find the mean deviation about the median of 28, 7, 22, 13, 25, 14, 9.
  3. 6For 1, 2, 3, 4, 10 find the mean deviation about the mean and about the median. Which is smaller?
Challenging
  1. 7Find the mean deviation of 5, 5, 5, 5 about the median.
  2. 8For 2, 3, 7, 12, 16 find the mean deviation about the mean and about the median.
Show answers (try them yourself first)
  1. 14
  2. 25
  3. 33
  4. 45.33
  5. 56.57
  6. 62.4 and 2.2; the median one
  7. 70
  8. 84.8 and 4.6

Mean deviation for grouped data

From daily life

A shop records how many customers buy 1, 2, 3 items. Instead of listing every customer, it keeps a frequency table. We need a way to find the spread from such a table.

Simple definition

For a table with values x and frequencies f (total N): MD = (sum of f·|x − mean|) ÷ N. For class intervals use the class mark (midpoint) as x. The mean is (sum of f·x) ÷ N.

Formula: MD = Σ f|x − mean| ÷ N

Rule 1: Discrete table — Find N, mean, then f·|x − mean|.

  • Easy: values 1, 2, 3, 4 with frequencies 2, 3, 3, 2: N = 10, mean 2.5, MD = 0.9.
  • Medium: values 10, 20, 30, 40 with frequencies 4, 6, 6, 4: N = 20, mean 25, MD = 9.
  • Exam-type: values 5, 10, 15 with frequencies 3, 4, 3: N = 10, mean 10, MD = 3.

Rule 2: Classes — Use class marks.

  • Easy: classes 0–10, 10–20, 20–30, 30–40 with frequencies 4, 6, 8, 2: class marks 5, 15, 25, 35; N = 20; mean 19; Σf|x − mean| = 160; MD = 8.
  • Medium: classes 0–20, 20–40, 40–60 with frequencies 6, 8, 6: class marks 10, 30, 50; mean 30; MD = 12.
  • Exam-type: Always work out the mean first, then the gaps.
Remember

Use frequency as the weight of each gap.

Common mistake

Dividing by the number of classes instead of N.

Memory clue

Weight the gaps by frequency.

Practice questions

Easy
  1. 1Find the mean deviation about the mean for values 1, 2, 3, 4 with frequencies 2, 3, 3, 2.
  2. 2Find the mean deviation about the mean for values 10, 20, 30, 40 with frequencies 4, 6, 6, 4.
  3. 3Find the mean deviation about the mean for values 5, 10, 15 with frequencies 3, 4, 3.
Medium
  1. 4Find the mean deviation about the mean for classes 0–10, 10–20, 20–30, 30–40 with frequencies 3, 5, 7, 5.
  2. 5Find the mean deviation about the mean for classes 0–20, 20–40, 40–60 with frequencies 6, 8, 6.
  3. 6Find the mean deviation about the mean for classes 10–20, 20–30, 30–40, 40–50 with frequencies 2, 5, 8, 5.
Challenging
  1. 7Find the mean deviation about the mean for values 2, 4, 6, 8, 10 with frequencies 1, 2, 4, 2, 1.
  2. 8Find the mean deviation about the mean for classes 0–4, 4–8, 8–12 with frequencies 5, 10, 5.
Show answers (try them yourself first)
  1. 10.9
  2. 29
  3. 33
  4. 48.6
  5. 512
  6. 67.6
  7. 71.6
  8. 82