Class 10 Mathematics – Statistics | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: A company surveys 500 customers about their age. The manager cannot read all 500 answers individually — instead, they calculate the mean (average age), mode (most common age group), and median (middle value) to understand their typical customer. Doctors use mean blood pressure, governments use average income, and teachers use class scores — statistics turns large piles of data into a few powerful, useful numbers.

📊 Topic 1: What Is Statistics?

Statistics is about collecting data.
We also organise and present data.
Then we analyse the data for meaning.
Data can be numbers, facts, or figures.
We use statistics in daily life.
Surveys, weather, and sports use it.

📊 Topic 2: Grouped Data

Grouped data uses class intervals.
A class interval is a range of values.
Example: 10–20, 20–30, 30–40.
The lower limit is the smaller value.
The upper limit is the larger value.
Class size (h) = upper − lower limit.
Frequency tells how many values fall in each class.

📊 Topic 3: Mean by Direct Method

Mean is the average of the data.
Find the midpoint (xᵢ) of each class.
Midpoint = (lower + upper) ÷ 2.
Multiply each midpoint by its frequency.
That gives fᵢxᵢ for each class.
Add all fᵢxᵢ values to get Σfᵢxᵢ.
Add all frequencies to get Σfᵢ.

Mean = Σfᵢxᵢ ÷ Σfᵢ

📊 Topic 4: Mean by Assumed Mean Method

Pick a value a as the assumed mean.
Usually pick the midpoint of a middle class.
Find dᵢ = xᵢ − a for each class.
Multiply: fᵢdᵢ for each class.
Sum up all fᵢdᵢ values.
This method reduces big calculations.

Mean = a + (Σfᵢdᵢ ÷ Σfᵢ)

📊 Topic 5: Mean by Step Deviation

This method simplifies even further.
Find uᵢ = (xᵢ − a) ÷ h.
Here h is the class size.
Multiply: fᵢuᵢ for each class.
Sum all fᵢuᵢ to get Σfᵢuᵢ.
Best when class size is the same throughout.

Mean = a + h × (Σfᵢuᵢ ÷ Σfᵢ)

📊 Topic 6: Median of Grouped Data

Median is the middle value of data.
Find n/2 where n = Σfᵢ.
Build a cumulative frequency column.
The class where cf first crosses n/2 is the median class.
l = lower limit of the median class.
cf = cumulative frequency before it.
f = frequency of the median class.

Median = l + [(n/2 − cf) ÷ f] × h

📊 Topic 7: Mode of Grouped Data

Mode is the most frequent value.
The class with the highest frequency is the modal class.
l = lower limit of the modal class.
f₁ = frequency of the modal class.
f₀ = frequency of the class before it.
f₂ = frequency of the class after it.

Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h

📊 Topic 8: Cumulative Frequency

Cumulative frequency is a running total.
Start from the first class frequency.
Add each next frequency to the total so far.
The last cumulative frequency = n (total).
We use it to find the median.
We also use it to draw an ogive.

📊 Topic 9: Ogive (Cumulative Frequency Curve)

An ogive is a cumulative frequency graph.
Plot upper class limits on the x-axis.
Plot cumulative frequencies on the y-axis.
Join the points with a smooth S-shaped curve.
A less than ogive rises upward.
A more than ogive falls downward.
Where the two ogives cross gives the median.

📊 Topic 10: Relation Between Mean, Median, and Mode

There is a useful empirical relation.
It connects all three averages.
This works for moderately skewed data.
If you know any two, find the third.
This is an approximate formula.

Mode = 3 Median − 2 Mean

🎨 Section 2 — Easy Illustrations

Picture 1: Frequency Table Visual

Each row shows one class interval, its frequency, and its midpoint.

Picture 2: Histogram (Bar Chart)

Bars touch each other in a histogram. Taller bars mean higher frequency.

Picture 3: Ogive (Cumulative Frequency Curve)

The S-shaped curve rises from zero to the total frequency. Where n/2 meets the curve gives the median.

Picture 4: Mean Formula Flowchart

Choose the method that makes your calculations easiest.

Picture 5: Median and Mode on a Number Line

Mean, median, and mode sit at different places. They are connected by an empirical formula.

✏️ Section 3 — Worked Examples

Example 1: Find the Mean (Direct Method)

Question: Find the mean of this data using the direct method.

Class	Frequency (fᵢ)	Midpoint (xᵢ)	fᵢxᵢ
10 – 20	4	15	60
20 – 30	7	25	175
30 – 40	6	35	210
40 – 50	3	45	135

1 Find midpoints: (10+20)/2 = 15, (20+30)/2 = 25, etc.

2 Multiply: fᵢ × xᵢ for each row.

3 Sum of fᵢxᵢ = 60 + 175 + 210 + 135 = 580.

4 Sum of fᵢ = 4 + 7 + 6 + 3 = 20.

5 Mean = Σfᵢxᵢ ÷ Σfᵢ = 580 ÷ 20 = 29.

Mean = 29

Example 2: Find the Median

Question: Find the median of this grouped data.

Class	Frequency (f)	Cumulative Frequency (cf)
0 – 10	5	5
10 – 20	8	13
20 – 30	12	25
30 – 40	9	34
40 – 50	6	40

1 Total n = Σf = 5+8+12+9+6 = 40.

2 n/2 = 40/2 = 20.

3 cf first crosses 20 at class 20–30 (cf = 25). This is the median class.

4 l = 20, cf (before) = 13, f = 12, h = 10.

5 Median = 20 + [(20 − 13) ÷ 12] × 10

6 = 20 + (7/12) × 10 = 20 + 5.83 = 25.83.

Median = 25.83

Example 3: Find the Mode

Question: Find the mode of this grouped data.

Class	Frequency
0 – 15	6
15 – 30	10
30 – 45	18 (highest)
45 – 60	8
60 – 75	3

1 Modal class = 30–45 (highest frequency = 18).

2 l = 30, f₁ = 18, f₀ = 10, f₂ = 8, h = 15.

3 Mode = 30 + [(18 − 10) ÷ (2×18 − 10 − 8)] × 15

4 = 30 + [8 ÷ (36 − 18)] × 15

5 = 30 + (8/18) × 15 = 30 + 6.67 = 36.67.

Mode = 36.67

Check: 2×18 − 10 − 8 = 18 ≠ 0 ✓

📊 Section 4 — Cheat Sheet

Term	What It Means	Key Fact
Statistics	Science of data	Collect, organise, analyse
Class Interval	Range of values	e.g. 10–20, 20–30
Frequency	How many in a class	Count of data points
Midpoint (xᵢ)	(Lower + Upper) ÷ 2	Centre of each class
Mean (Direct)	Σfᵢxᵢ ÷ Σfᵢ	Average of data
Assumed Mean	a + Σfᵢdᵢ / Σfᵢ	dᵢ = xᵢ − a
Step Deviation	a + h × Σfᵢuᵢ / Σfᵢ	uᵢ = (xᵢ − a)/h
Median	Middle value	l + [(n/2−cf)/f]×h
Mode	Most frequent value	l+[(f₁−f₀)/(2f₁−f₀−f₂)]×h
Ogive	Cumulative frequency graph	S-shaped curve
Cumulative Freq	Running total of f	Used for median
Empirical Relation	Connects mean, median, mode	Mode = 3Med − 2Mean

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. The midpoint of a class interval is (lower + upper) ÷ _________.
2. In the assumed mean method, dᵢ = xᵢ − _________.
3. The class with the highest frequency is the _________ class.
4. Cumulative frequency is a _________ total of frequencies.
5. The empirical relation is Mode = 3 Median − _________ Mean.

✓ B. True or False

1. Mean can be found using three different methods. _______
2. The median class is the class with highest frequency. _______
3. An ogive is always an S-shaped curve. _______
4. Mode = 3 Mean − 2 Median. _______
5. Class size h = upper limit − lower limit. _______

✏️ C. Solve These

1. Find the mean of the following data using the direct method:

Classes: 5–15, 15–25, 25–35, 35–45. Frequencies: 3, 8, 5, 4.

Show your work here

2. Find the mode: Classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 4, 7, 15, 9, 5.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Mean	a. Highest frequency class
2. Modal class	b. S-shaped curve
3. Ogive	c. Running total
4. Cumulative frequency	d. Σfᵢxᵢ ÷ Σfᵢ

🎨 E. Draw the Ogive

Draw a less-than ogive for the data: Classes 0–10, 10–20, 20–30, 30–40. Frequencies: 3, 7, 12, 8.

Draw your ogive here

✅ Detailed Solutions

Solution to Q1: Mean from Grouped Data

Question: Find the mean of the following frequency distribution: Classes 5–15, 15–25, 25–35, 35–45 with frequencies 3, 8, 5, 4.

1 Find the midpoint (x) of each class: 10, 20, 30, 40.

2 Calculate f × x for each class: 3×10 = 30, 8×20 = 160, 5×30 = 150, 4×40 = 160.

3 Σfx = 30 + 160 + 150 + 160 = 500.

4 Σf = 3 + 8 + 5 + 4 = 20.

5 Mean = Σfx ÷ Σf = 500 ÷ 20 = 25.

∴ Mean = 25

Solution to Q2: Mode from Grouped Data

Question: Find the mode of: Classes 0–10(4), 10–20(7), 20–30(15), 30–40(9), 40–50(5).

1 The modal class is the class with the highest frequency = 20–30 (f = 15).

2 l = 20 (lower limit), h = 10 (class width), f₁ = 15, f₀ = 7, f₂ = 9.

3 Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h.

4 = 20 + [(15 − 7) ÷ (30 − 7 − 9)] × 10 = 20 + (8/14) × 10.

5 = 20 + 80/14 = 20 + 5.71 = 25.71.

∴ Mode = 25.71

🃏 Section 6 — Revision Flashcards

👉

📊

What is the direct method formula for mean?

Mean = Σfᵢxᵢ ÷ Σfᵢ

Add all f×x, then divide by total frequency.

🔎

How do you find the median class?

Find n/2. Build cumulative frequency. The class where cf first crosses n/2 is the median class.

📈

What is the median formula for grouped data?

Median = l + [(n/2 − cf) ÷ f] × h

l = lower limit, cf = cumulative freq before, f = freq of median class, h = class size.

🔗

What is the mode formula for grouped data?

Mode = l + [(f₁−f₀) ÷ (2f₁−f₀−f₂)] × h

f₁ = modal class freq, f₀ = class before, f₂ = class after.

❓

What is cumulative frequency?

A running total of frequencies. Each row adds its frequency to the sum of all rows above it.

💡

What is an ogive?

A graph of cumulative frequency against upper class limits. It makes an S-shaped curve. Used to find the median.

🔧

What is the empirical relation between mean, median, and mode?