Class 10 - Statistics Formula Sheet

1. Mean of Grouped Data

Direct Method

Mean = S(f?x?) / Sf?
Where:
x? = class mark
f? = frequency

Assumed Mean Method

Mean = A + (Sf?d? / Sf?)
Where:
A = assumed mean
d? = x? - A

Step Deviation Method

Mean = A + h(Sf?u? / Sf?)
Where:
u? = (x? - A)/h
h = class width

2. Mode of Grouped Data

Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] � h

Where:
l = lower limit of modal class
f1 = frequency of modal class
f0 = frequency of class preceding modal class
f2 = frequency of class succeeding modal class
h = class size

Modal Class: Class with highest frequency

3. Median of Grouped Data

Median = l + [(n/2 - cf) / f] � h

Where:
l = lower limit of median class
n = total frequency (Sf?)
cf = cumulative frequency of class preceding median class
f = frequency of median class
h = class size

Median Class: First class where cumulative frequency = n/2

4. Cumulative Frequency

Types of Cumulative Frequency:

Less than type: Upper limits are considered
More than type: Lower limits are considered

Cumulative Frequency (CF) up to i?? class = f1 + f2 + ... + f?

5. Empirical Relationship

Mode = 3Median - 2Mean

This relationship holds approximately for moderately skewed distributions

6. Ogive (Cumulative Frequency Curve)

Less than Ogive

� Plot upper limits vs cumulative frequency
� Smooth curve joining points
� Used to find median

More than Ogive

� Plot lower limits vs cumulative frequency
� Smooth curve joining points
� Intersection with less than ogive gives median

7. Class Mark Formula

Class Mark = (Lower Limit + Upper Limit) / 2

8. Finding Median Graphically

Draw less than ogive
Locate n/2 on y-axis (cumulative frequency)
Draw horizontal line to meet ogive
From meeting point, draw vertical line to x-axis
Point on x-axis is median

9. Quick Reference Table

Measure	Formula	When to Use
Mean (Direct)	S(f?x?)/Sf?	Small numbers, no common factor
Mean (Assumed)	A + Sf?d?/Sf?	Large numbers
Mean (Step Deviation)	A + h(Sf?u?/Sf?)	Equal class intervals
Mode	l + [(f1-f0)/(2f1-f0-f2)]�h	Highest frequency class
Median	l + [(n/2-cf)/f]�h	Middle value position

10. Example Problems (NCERT Style)

Problem 1: Find mean of:
Class: 0-10, 10-20, 20-30, 30-40, 40-50
Frequency: 5, 10, 15, 7, 3

Solution (Direct Method):

Class	f?	x?	f?x?
0-10	5	5	25
10-20	10	15	150
20-30	15	25	375
30-40	7	35	245
40-50	3	45	135
Total	40		930

Mean = 930/40 = 23.25

Problem 2: Find mode of:
Class: 0-10, 10-20, 20-30, 30-40, 40-50
Frequency: 5, 8, 12, 6, 4

Solution:
Modal class = 20-30 (highest frequency 12)
l = 20, f1 = 12, f0 = 8, f2 = 6, h = 10
Mode = 20 + [(12-8)/(2�12 - 8 - 6)] � 10
= 20 + [4/(24-14)] � 10 = 20 + (4/10)�10 = 20 + 4 = 24

Problem 3: Find median of:
Class: 0-10, 10-20, 20-30, 30-40, 40-50
Frequency: 5, 8, 20, 15, 7

Solution:
n = 55, n/2 = 27.5
Median class = 20-30 (cf up to 10-20 = 13, cf up to 20-30 = 33)
l = 20, cf = 13, f = 20, h = 10
Median = 20 + [(27.5-13)/20] � 10
= 20 + (14.5/20)�10 = 20 + 7.25 = 27.25

11. Important Points to Remember

Class size (h) = Upper limit - Lower limit
Class mark = Midpoint of class interval
Cumulative frequency is running total
For median: n/2, not n
Modal class has highest frequency
Median class: First class where CF = n/2
Empirical relation: Mode � 3Median - 2Mean
Ogive gives median graphically
All formulas require continuous classes

12. Step Deviation Method Example

Class	f?	x?	u?=(x?-25)/10	f?u?
0-10	5	5	-2	-10
10-20	10	15	-1	-10
20-30	15	25	0	0
30-40	7	35	1	7
40-50	3	45	2	6
Total	40			-7

A = 25, h = 10, Sf? = 40, Sf?u? = -7
Mean = 25 + 10�(-7/40) = 25 - 1.75 = 23.25