Complete guide — graphical representation (bar graphs, histograms, frequency polygons) and measures of central tendency (mean, median, mode). Chapter 14 · 5 Marks.
📌 Chapter 14 · 5 Marks
Bar Graph
Histogram
Frequency Polygon
Mean · Median · Mode
50 Practice Qs
Section 14.1 — Graphical Representation
Bar Graphs, Histograms & Frequency Polygons
The three main ways to display statistical data visually. Know when to use each one.
Graph Type
Used When
Key Feature
Bars/Lines
Bar Graph
Discrete / categorical data
Gaps between bars
Bars of equal width
Histogram
Continuous data in class intervals
No gaps between bars
Width = class size
Frequency Polygon
Comparing distributions
Line graph on histogram midpoints
Connect midpoints of tops
Bar graph — discrete categorical data, gaps between bars, all bars same width
Histogram — continuous data, NO gaps between bars. Frequency polygon — connect midpoints of bar tops
Section 14.2
Mean (Arithmetic Average)
The mean is the sum of all observations divided by the number of observations. Most common measure of central tendency.
📐 Mean Formulas
Mean (x̄) = Σxᵢ / n
For grouped data: x̄ = Σfᵢxᵢ / Σfᵢ
where xᵢ = value (or class midpoint), fᵢ = frequency, n = Σfᵢ = total observations
Example 1 · 2M
Find mean of: 8, 3, 7, 12, 5, 9, 11, 6, 4, 15.
1
Σx = 8+3+7+12+5+9+11+6+4+15 = 80
2
n = 10. Mean = 80/10 = 8
Mean = 8
Example 2 · 3M
Find mean for grouped data:
Marks (xᵢ)
Frequency (fᵢ)
fᵢxᵢ
10
3
30
20
5
100
30
8
240
40
4
160
Total
20
530
Mean = Σfᵢxᵢ / Σfᵢ = 530 / 20 = 26.5
Example 3 · 2M
Mean of 5 numbers is 18. A 6th number is added; new mean = 20. Find the 6th number.
1
Sum of 5 = 5×18 = 90
2
Sum of 6 = 6×20 = 120
3
6th number = 120−90 = 30
The 6th number is 30.
Section 14.3
Median
The median is the middle value when data is arranged in ascending or descending order. It splits the data into two equal halves.
📐 Median Formulas
Odd n: M = [(n+1)/2]th term Even n: M = avg of (n/2)th and (n/2+1)th
Always arrange data in order first. The median is NOT affected by extreme values (unlike mean).
Odd n · 2M
Find median of: 3, 7, 5, 1, 9, 4, 6.
1
Arrange: 1, 3, 4, 5, 6, 7, 9
2
n=7 (odd). Middle = (7+1)/2 = 4th term = 5
Median = 5
Even n · 2M
Find median of: 8, 4, 12, 3, 7, 10.
1
Arrange: 3, 4, 7, 8, 10, 12
2
n=6 (even). Median = (3rd+4th)/2 = (7+8)/2 = 7.5
Median = 7.5
Section 14.4
Mode
The mode is the value that appears most frequently in the dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).
Mode = the observation with the highest frequency.
For grouped data (histogram): mode is the class with the highest frequency bar (the modal class).
ALWAYS arrange data in ascending order before finding median. Forgetting this gives the wrong middle value.
2
Mean × n = Sum
If mean and n are given, multiply to get sum. When observations are added/removed, update sum and n separately, then divide.
3
Histogram: no gaps, equal width
Bars touch each other. If class widths differ, area of bar (not height) represents frequency. CBSE mostly uses equal widths.
4
Frequency polygon: join midpoints
Plot midpoint of each class vs frequency. Join with straight lines. Extend to zero frequency at both ends (ghost classes).
5
Mode = 3M − 2x̄
Empirical formula: Mode ≈ 3×Median − 2×Mean. Given any two, find the third. Appears in CBSE 2M questions.
6
Median unaffected by extremes
If data has outliers (very large/small values), median is better than mean. A billionaire in a village doesn't change the median income but greatly changes the mean.
7
Bar graph vs histogram
Bar graph: discrete data, gaps between bars. Histogram: continuous grouped data, NO gaps. CBSE asks you to identify and draw both.
8
Grouped mean: use midpoints
For class intervals (e.g., 10–20), the value xᵢ is the midpoint = (10+20)/2 = 15. Then x̄ = Σfᵢxᵢ/Σfᵢ.
CBSE Practice
50 Practice Questions
MCQ · 1M · 2M · 3M · 5M · Case-Based — all with solutions.