CBSE Class 9 Maths — Statistics

Chapter Overview & Weightage

Statistics in Class 9 introduces students to data organisation, graphical representation, and measures of central tendency. These skills are foundational for science and social science. In CBSE SA exams, this chapter consistently carries 8–10 marks.

Year	Marks	Question Types
2023	10	2 MCQ + 1 SA + 1 LA (finding mean/median)
2022	8	1 MCQ + 2 SA + 1 LA (drawing histogram)
2021	10	2 SA + 1 LA
2020	8	1 SA + 1 LA

Mean, median, and mode from grouped data, and drawing histograms/frequency polygons, are the most tested topics. Expect at least one 4–5 mark question requiring full calculation of mean by direct, deviation, or step deviation method.

Key Concepts You Must Know

Data types:

Raw data: Data in the original, unorganised form
Frequency: How many times a value occurs
Class interval: Range of values in a group (e.g., 10–20, 20–30)
Class width/size: Difference between upper and lower limits (e.g., 10 for 10–20)
Class mark/midpoint: $(upper + lower)/2$

Measures of central tendency:

Mean: Average — sum of all values ÷ number of values
Median: The middle value when data is arranged in order
Mode: The value that appears most frequently

Graphical representations:

Bar graph: For discrete, categorical data (bars don’t touch)
Histogram: For continuous data in class intervals (bars touch — no gap)
Frequency polygon: Connect midpoints of tops of histogram bars with straight lines
Ogive (cumulative frequency curve): Useful for finding median graphically

Important Formulas

\text{Mean} (\bar{x}) = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}

Median:

Odd $n$ : Middle value = $\left(\frac{n+1}{2}\right)^{th}$ term
Even $n$ : Average of $\left(\frac{n}{2}\right)^{th}$ and $\left(\frac{n}{2}+1\right)^{th}$ terms

Mode: Value with highest frequency

Direct Method:

\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

Assumed Mean (Deviation) Method:

\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}, \quad d_i = x_i - a

Step Deviation Method:

\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h, \quad u_i = \frac{x_i - a}{h}

Where $a$ = assumed mean, $h$ = class width, $x_i$ = class mark

Solved Previous Year Questions

PYQ 1 — 2023 CBSE

Q: Find the mean of the following data:

Class	0–10	10–20	20–30	30–40	40–50
Frequency	5	8	15	7	5

Solution:

Class	$x_i$ (midpoint)	$f_i$	$f_i x_i$
0–10	5	5	25
10–20	15	8	120
20–30	25	15	375
30–40	35	7	245
40–50	45	5	225
Total		40	990

\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{990}{40} = 24.75

Mean = 24.75

PYQ 2 — 2022 CBSE

Q: The following marks (out of 50) were obtained by 30 students. Find the median: 30, 19, 25, 30, 27, 36, 28, 33, 27, 28, 30, 28, 19, 35, 27, 22, 27, 28, 31, 32, 22, 25, 35, 27, 28, 30, 22, 27, 20, 25

Arrange in ascending order and find the 15th and 16th values (since $n = 30$ , even):

After arranging: 19, 19, 20, 22, 22, 22, 25, 25, 25, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 30, 30, 30, 30, 31, 32, 33, 35, 35, 36

15th value = 27, 16th value = 28

\text{Median} = \frac{27 + 28}{2} = 27.5

PYQ 3 — 2021 CBSE

Q: Draw a histogram for the data:

Age (years)	5–10	10–15	15–20	20–25	25–30
Number	6	11	21	23	14

Steps: Draw x-axis with class intervals (5–10, 10–15, …), y-axis with frequency. Draw rectangles with no gaps between them. Height of each rectangle = frequency. Mark equal class widths.

Difficulty Distribution

Difficulty	%	Topics
Easy	35%	Mean of ungrouped data, mode, reading bar graphs
Medium	45%	Mean from grouped data, constructing histogram, finding median
Hard	20%	Combining datasets, using step-deviation method, drawing ogive

Expert Strategy

Master the class mark calculation. In every grouped data problem, you need the midpoint of each class. For class 20–30: midpoint = 25. Get this right before proceeding.

Choose the right method for mean:

Direct method: Use when numbers are small and arithmetic is easy
Assumed mean method: When numbers are large (like 450–500 range)
Step deviation: Most efficient when class widths are equal (use $h$ = class width)

For the step deviation method, always choose $a$ (assumed mean) as the class mark closest to the “middle” of your distribution. This minimises the $u_i$ values and makes arithmetic simpler. A wrong choice of $a$ gives the same answer but harder calculations.

Histogram vs bar graph: A histogram is for continuous data with class intervals — bars must touch. A bar graph is for discrete categories — bars can be separated. Drawing a histogram with gaps between bars is wrong and loses marks.

Common Traps

Trap 1: Using upper or lower limit as class mark instead of midpoint. For class 10–20, the class mark is $(10+20)/2 = 15$ , not 10 or 20. Using the wrong value changes the mean calculation completely.

Trap 2: Forgetting to add the $a$ (assumed mean) back at the end in the deviation method. The formula is $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$ . Students often calculate $\frac{\sum f_i d_i}{\sum f_i}$ correctly but forget to add $a$ , giving an answer close to 0 instead of the actual mean.

Trap 3: In frequency polygon, students connect the first class mark to the last class mark without extending to the x-axis. Frequency polygons should start and end at the x-axis — add imaginary classes with frequency 0 at both ends and connect to those midpoints.

Trap 4: Confusing mode of grouped data with ungrouped data. For ungrouped data, mode = most frequent value. For grouped data at Class 9 level, mode = class with highest frequency (the modal class). More precise calculation of mode from grouped data is in Class 10.