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CBSE Class 10 · Mathematics

Statistics

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Statistics” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Mean of Grouped Data

The mean of grouped data uses class midpoints as stand-ins: x_i = (low+high)/2, then x̄ = Σ f_i x_i / Σ f_i (direct method as taught). A class 0–10 with f = 4 has midpoint 5, not 10. The mean is a balance-point of those stand-ins, not a person in the list.

Using the upper limit as the midpoint is a miss.

Figure. Grouped mean uses the class mark, the mid-point of each interval. With frequencies 4, 7, 12, 6 the marks 5, 15, 25, 35 give Σfx = 635 and mean 635/29 ≈ 21.9. Height tracks frequency, not the mark.

How it works

  1. Write midpoints for each classThe stand-ins.
  2. Form Σ f x and Σ fThe sums.
  3. DivideThe grouped mean.

Three classes

Classes 0–10, 10–20, 20–30 with frequencies 4, 6, 5. Find the mean (midpoints).

  • Midpoints5, 15, 25
  • Σ f x4×5 + 6×15 + 5×25 = 20+90+125 = 235
  • Σ f = 15, mean235/15 = 47/3

Pro tip. Midpoint stand-ins; 47/3 is the balance, not a student.

2Mode of Grouped Data

The mode of grouped data sits in the modal class (highest frequency) and then uses the school formula with the neighbouring frequencies, as taught. Mode is the peak-class story, not “the midpoint of the tallest bar” unless the formula reduced to that. A tie for tallest is a careful case, not a shrug.

Calling the highest score in the raw list the mode of grouped data skips the class.

Figure. Modal class is the tallest bar, 20–30. The two dashed joins from that bar to its neighbours meet above 24.5, which is 20 + 10×5/11. Frequencies are not to a shared newton scale with any other figure; only these four heights are compared.

How it works

  1. Find the class with largest fThe modal class.
  2. Apply the taught mode formula with neighboursThe mode.
  3. Read it as a peak-estimate, not a named studentGrouped mode.

3Median of Grouped Data

The median of grouped data uses the median class (the class where the cumulative frequency crosses n/2) and the school formula with l, n, cf, f, h as taught. Median is a middle-of-the-list estimate, not the mean. For n = 15, n/2 = 7.5; find which class holds that count.

Using the mean’s Σ f x as if it were the median is the heading-steal.

Figure. Less-than ogive of the same four classes: cumulative 0, 4, 11, 23, 29. A dashed rule at n/2 = 14.5 meets the curve above 22.9, the median class mark from the 20–30 interval.

How it works

  1. Form cf and find n/2The middle-count.
  2. Name the median classWhere cf crosses n/2.
  3. Apply the taught median formulaThe grouped median.

4A definition is a test you can run

Mean / mode / median (grouped) is a test: midpoints and Σ f x, or modal class, or cf crossing n/2. If you only say “average”, you have a heading.

A bar chart with no f-list is not the test.

Figure. A statistics sentence is a test: the classes and frequencies you named must add to n before any mean is honest.

How it works

  1. Name which average and the class-listThe object.
  2. Run the matching school writeThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given is the class table. The unknown is mean, mode, or median. Copy the frequencies before you invent a midpoint 10 for 0–10.

Merging 10–20 into 0–10 because the drawing was tight is a silent given.

Figure. Name the frequencies and the class marks before you ask for the mean. The unknown sits after the given pile, not in front of it.

How it works

  1. Copy classes and fThe given.
  2. Name which averageThe unknown.
  3. Then midpoints or cfGiven first.

6One worked case is enough at this class

One small table is enough. A page of the same 4-6-5 clone does not add a new mode.

Ten twins of 235/15 are still one idea.

Figure. One grouped table is enough to run mean, mode, and median at this class. A second table is a new question, not a required check.

How it works

  1. Run one mean or one median-classThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A crowd-photo is a setting. The maths is a class table. Faces are not frequencies until counted into bins.

A stadium still is not Σ f x.

Figure. The same height with a swapped name is a different picture. Twelve visits is not twelve days. Read the axis noun before you read the number.

How it works

  1. Keep the crowd as a storyA setting.
  2. Write the class tableThe maths.
  3. Then mean, mode, or medianDo not swap.

8Check by the opposite action or the opposite test

Check a mean by recomputing Σ f x; check a median class by walking cf again. A mode that sits outside the modal class fails the opposite test.

Comparing mean to median as “they should be equal” is not required.

Figure. The opposite test of a mean is multiply back by n. 635 ÷ 29 ≈ 21.9, and 21.9 × 29 returns the same 635.

How it works

  1. State 47/3 or the modal class 10–20The claim.
  2. Rebuild the sums or the cfThe check.
  3. Mend if they disagreeHonest.
Midpoint of 0–10 is
  1. 5
  2. 10
  3. 0

Average of the ends; not the upper limit.

Notes

  • The official chapter title is “Statistics”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • x̄ = Σ f x / Σ f with x = class midpoint
  • median class: cf crosses n/2

Recap

Hold these pegs from the official chapter “Statistics”. The wording is ExamMaster’s teaching, not a textbook recap.

Mean of Grouped Data
The mean of grouped data uses class midpoints as stand-ins: x_i = (low+high)/2, then x̄ = Σ f_i x_i / Σ f_i (direct method as taught).
Mode of Grouped Data
The mode of grouped data sits in the modal class (highest frequency) and then uses the school formula with the neighbouring frequencies, as taught.
Median of Grouped Data
The median of grouped data uses the median class (the class where the cumulative frequency crosses n/2) and the school formula with l, n, cf, f, h as taught.
A definition is a test you can run
Mean / mode / median (grouped) is a test: midpoints and Σ f x, or modal class, or cf crossing n/2.
Name the given before the unknown
The given is the class table. The unknown is mean, mode, or median. Copy the frequencies before you invent a midpoint 10 for 0–10.
One worked case is enough at this class
One small table is enough. A page of the same 4-6-5 clone does not add a new mode.

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