Frequency Distribution – పౌనఃపున్య పంపిణీ.

పెద్ద మొత్తంలో ఉన్న Raw Dataను నేరుగా పరిశీలించడం కష్టం. అందువల్ల observationsను values లేదా class intervals ప్రకారం arrange చేసి, ప్రతి value/class ఎన్ని సార్లు occur అయిందో చూపిస్తాము. దీనినే Frequency Distribution అంటారు.

APPSC Focus:

Raw Data & Array
Frequency
Frequency Distribution
Ungrouped & Grouped Data
Discrete & Continuous Frequency Distribution
Class Interval
Lower & Upper Class Limits
Class Boundaries / Real Limits
Class Width / Class Size
Class Midpoint / Class Mark
Inclusive & Exclusive Method
Relative Frequency
Percentage Frequency
Cumulative Frequency
Less-than & More-than Cumulative Frequency
Numerical Problems & Shortcuts

1. Raw Data – ముడి దత్తాంశం

Dataను collect చేసిన original sequenceలో, ఇంకా classify లేదా process చేయకముందు ఉన్న dataను Raw Data అంటారు.

Example:

12, 18, 15, 12, 20, 18, 14, 15, 18, 12

Raw Data:

Data recorded in the sequence in which they are collected and before they are processed.

2. Array

Raw dataను ascending లేదా descending orderలో arrange చేస్తే దానిని Array అంటారు.

Raw Data:
12, 18, 15, 12, 20, 18, 14, 15, 18, 12

Ascending Array:
12, 12, 12, 14, 15, 15, 18, 18, 18, 20

3. Frequency – పౌనఃపున్యం

ఒక value లేదా class data setలో ఎన్ని సార్లు occur అయిందో ఆ సంఖ్యను Frequency అంటారు.

Data:

2, 3, 3, 4, 4, 4, 5

Frequency of 2 = 1
Frequency of 3 = 2
Frequency of 4 = 3
Frequency of 5 = 1

⚡ Memory:

Frequency = Number of Occurrences

4. Frequency Distribution – Definition

A Frequency Distribution is a table that lists the values, categories or classes and the number of observations belonging to each value or class.

Quantitative dataలో raw observationsను different classesలో classify చేసి వాటి corresponding class frequenciesను చూపడం frequency distribution యొక్క ప్రధాన ఉద్దేశ్యం.

5. Simple Frequency Distribution

The number of books read by 15 students is:

1, 2, 3, 2, 4, 1, 2, 3, 2, 5, 4, 3, 2, 1, 4

Books (X)	Tally	Frequency (f)
1	|||	3
2	|||||	5
3	|||	3
4	|||	3
5	|	1
Total		15

⚡ First Check:

Always verify:

Σf = Total Number of Observations

6. Ungrouped Data vs Grouped Data

Ungrouped Data	Grouped Data
Individual observations are available	Observations are classified into classes
Original detail is retained	Data are summarized
Suitable for relatively small data	Useful for large data sets
Example: 12, 15, 18, 20	Example: 10–20, 20–30, 30–40

Data presented in a frequency distribution table are called Grouped Data.

7. Discrete Frequency Distribution

Variable distinct or countable values తీసుకున్నప్పుడు ప్రతి valueకు corresponding frequency చూపించే distributionను Discrete Frequency Distribution అంటారు.

Number of Children (X)	Families (f)
0	4
1	8
2	15
3	10
4	3

8. Continuous Frequency Distribution

Observationsను class intervalsలో group చేస్తే continuous frequency distribution ఏర్పడుతుంది.

Marks	Frequency
0–10	3
10–20	7
20–30	12
30–40	8
40–50	5

9. Class Interval

Frequency distributionలో observationsను group చేయడానికి ఉపయోగించే intervalను Class Interval అంటారు.

Example:

10–20, 20–30, 30–40, 40–50

Each of the above is a class interval.

10. Class Limits

Each classకు two ends ఉంటాయి:

Lower Class Limit (LCL) → Lower end of the class

Upper Class Limit (UCL) → Upper end of the class

For class 60–70:

Lower Class Limit = 60
Upper Class Limit = 70

APPSC Trap:

Class Limit and Class Boundary are not necessarily the same.

11. Class Width / Class Size

A class యొక్క upper boundary మరియు lower boundary మధ్య differenceను Class Width లేదా Class Size అంటారు.

Class Width = Upper Class Boundary − Lower Class Boundary

Equal continuous intervals such as 10–20, 20–30లో:

Width = 20 − 10 = 10

Class Width = 10

12. Class Midpoint / Class Mark

Class యొక్క central valueను Class Midpoint లేదా Class Mark అంటారు.

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

Find the class mark of 20–30.

(20 + 30) / 2 = 25

Answer: 25

⚡ Shortcut:

Class Mark = Average of the two class limits.

13. Class Boundaries / Real Class Limits

Adjacent classes మధ్య gapను remove చేసి continuous classesగా మార్చడానికి Class Boundaries ఉపయోగిస్తారు.

Uploaded textbook ప్రకారం, one class upper limit మరియు next class lower limit యొక్క midpoint common class boundary అవుతుంది.

Classes:

10–19
20–29

19 మరియు 20 మధ్య midpoint:

(19 + 20) / 2 = 19.5

Therefore boundaries become:

9.5–19.5
19.5–29.5

14. Boundary Correction Shortcut

Successive class limits మధ్య gap:

Gap = Next Lower Limit − Previous Upper Limit

Correction factor:

Correction = Gap / 2

Classes are 20–29 and 30–39.

Gap = 30 − 29 = 1

Correction = 1/2 = 0.5

Therefore:

20–29 → 19.5–29.5
30–39 → 29.5–39.5

⚡ Integer-Class Shortcut:

For classes such as 10–19, 20–29, 30–39:

Lower Boundary = Lower Limit − 0.5
Upper Boundary = Upper Limit + 0.5

15. Inclusive Method

Both lower and upper class limits are treated as belonging to a class in the Inclusive Method.

Examples:

10–19
20–29
30–39
40–49

For example, 19 belongs to the first class and 20 belongs to the second class.

16. Exclusive Method

In the Exclusive Method, lower limit is included while upper limit is excluded and becomes the lower limit of the next class.

0–10
10–20
20–30
30–40

For class 10–20:

10 is included.
20 goes to the next class 20–30.

Important Trap:

In exclusive classes:

30–40 does not include 40.

40 belongs to 40–50.

17. Inclusive vs Exclusive Method

Inclusive	Exclusive
10–19, 20–29	10–20, 20–30
Upper limit included	Upper limit excluded
Often used with discrete observations	Suitable for continuous classification
May require boundary conversion	Classes are already continuous

18. Relative Frequency

A class frequencyని total observationsతో divide చేస్తే Relative Frequency వస్తుంది.

Relative Frequency = f / N

where:

f = Frequency of the class
N = Total number of observations = Σf

Numerical 1 – Relative Frequency

A class has frequency 15 and total frequency is 60. Find its relative frequency.

15 / 60 = 0.25

Answer: 0.25

19. Percentage Frequency

Percentage Frequency = Relative Frequency × 100

or directly:

Percentage = (f / N) × 100

Frequency = 15, Total = 60.

(15/60) × 100 = 25%

Answer: 25%

⚡ Two Mandatory Checks:

Σ Relative Frequencies = 1

Σ Percentages = 100%

Small rounding differences may occur when individual values are rounded.

20. Complete Relative Frequency Numerical

Class	f	Relative Frequency	Percentage
0–10	5	5/50 = 0.10	10%
10–20	10	10/50 = 0.20	20%
20–30	20	20/50 = 0.40	40%
30–40	10	0.20	20%
40–50	5	0.10	10%
Total	50	1.00	100%

21. Cumulative Frequency – సంచిత పౌనఃపున్యం

Frequenciesను successively add చేస్తూ పొందే frequencyను Cumulative Frequency (c.f.) అంటారు.

Cumulative frequency of a class = frequency of that class + frequencies of preceding classes.

Class	Frequency (f)	Cumulative Frequency
0–10	5	5
10–20	8	13
20–30	12	25
30–40	10	35
40–50	5	40

⚡ Most Important Check:

Last Less-than Cumulative Frequency
= Total Frequency N

22. Less-Than Cumulative Frequency

Starting from the first class, frequenciesను successively add చేస్తే Less-than Cumulative Frequency వస్తుంది.

Class	f	Less-than c.f.
0–10	4	4
10–20	6	10
20–30	10	20
30–40	8	28
40–50	2	30

Therefore:

Less than 10 → 4
Less than 20 → 10
Less than 30 → 20
Less than 40 → 28
Less than 50 → 30

23. More-Than Cumulative Frequency

More-than cumulative frequency total frequencyతో ప్రారంభమై class frequenciesను successively subtract చేస్తుంది.

For the previous table, N = 30.

More Than	c.f.
More than 0	30
More than 10	26
More than 20	20
More than 30	10
More than 40	2
More than 50	0

⚡ Memory:

Less-than c.f. → Add downward

More-than c.f. → Start with N and subtract downward

24. Recover Frequency from Cumulative Frequency

APPSCలో cumulative frequency ఇచ్చి ordinary frequency అడగవచ్చు.

Frequency = Current C.F. − Previous C.F.

Less-than cumulative frequencies are:

Less than 10 = 5
Less than 20 = 13
Less than 30 = 25
Less than 40 = 40

Class frequencies:

0–10 → 5

10–20 → 13 − 5 = 8

20–30 → 25 − 13 = 12

30–40 → 40 − 25 = 15

⚡ PYQ-Type Shortcut:

Cumulative → Ordinary Frequency
= Successive Difference

25. Cumulative Relative Frequency

Cumulative Relative Frequency = Cumulative Frequency / Total Observations

Cumulative Percentage = Cumulative Relative Frequency × 100

Cumulative frequency = 30 and N = 50.

Cumulative Relative Frequency = 30/50 = 0.60

Cumulative Percentage = 60%

26. Constructing a Grouped Frequency Distribution

Marks of 20 students:

12, 18, 25, 31, 29, 16, 22, 35, 41, 27, 19, 33, 24, 38, 45, 21, 28, 32, 17, 26

Using class width 10:

Class	Frequency
10–20	5
20–30	8
30–40	5
40–50	2
Total	20

Exclusive Method Reminder:

10–20 means 10 ≤ X < 20.
Therefore 20 would belong to 20–30, not 10–20.

27. Find Missing Frequency

The total frequency is 50. Find x.

Class	Frequency
0–10	8
10–20	12
20–30	x
30–40	15
40–50	5

8 + 12 + x + 15 + 5 = 50

40 + x = 50

x = 10

Answer: 10

⚡ Shortcut:

Missing Frequency = Total − Sum of Known Frequencies

28. Find Frequency from Percentage

Total observations = 200.
A class contains 15% of observations.
Find its frequency.

f = 15% of 200

= 15/100 × 200 = 30

Answer: 30

29. Find Total Frequency from Relative Frequency

A class has frequency 18 and relative frequency 0.15. Find total observations.

Relative Frequency = f/N

0.15 = 18/N

N = 18/0.15 = 120

Answer: 120

30. APPSC High-Yield Numerical Shortcuts

Asked	Shortcut / Formula
Total Frequency	Σf
Relative Frequency	f/N
Percentage	(f/N) × 100
Frequency from Percentage	(Percentage/100) × N
Class Mark	(L + U)/2
Class Width	Upper Boundary − Lower Boundary
Less-than C.F.	Successive Addition
Frequency from Less-than C.F.	Successive Difference
More-than C.F.	Start with N and successively subtract
Missing Frequency	N − Sum of Known Frequencies

31. APPSC Exam Traps

Trap 1:
Frequency = Value of variable → FALSE
Frequency = Number of times a value/class occurs.

Trap 2:
Class Mark = U − L → FALSE
Class Mark = (L + U)/2.

Trap 3:
Class Width = (L + U)/2 → FALSE
Class Width = Upper Boundary − Lower Boundary.

Trap 4:
Σ Relative Frequency = 100 → FALSE
It is 1.

Trap 5:
Σ Percentage Frequency = 1 → FALSE
It is 100%.

Trap 6:
Last less-than cumulative frequency = last class frequency → FALSE
It equals total frequency N.

Trap 7:
10–20 in exclusive method includes 20 → FALSE.

Trap 8:
Class limits and class boundaries are always identical → FALSE.

APPSC Objective Questions

1. The number of observations belonging to a class is called:

A) Class mark
B) Frequency
C) Boundary
D) Range

Answer: B – Frequency

2. Data presented in a frequency distribution are called:

A) Raw data only
B) Grouped data
C) Primary data only
D) Secondary data only

Answer: B – Grouped Data

3. The lower and upper ends of a class are called:

A) Frequencies
B) Class limits
C) Class marks
D) Cumulative frequencies

Answer: B – Class Limits

4. The class mark of 30–40 is:

A) 10
B) 30
C) 35
D) 40

Answer: C – 35

5. If the class boundaries are 19.5 and 29.5, class width is:

A) 5
B) 9
C) 10
D) 49

Answer: C – 10

6. If f = 20 and N = 80, relative frequency is:

A) 0.20
B) 0.25
C) 25
D) 4

20/80 = 0.25

Answer: B – 0.25

7. In Question 6, percentage frequency is:

A) 0.25%
B) 2.5%
C) 20%
D) 25%

Answer: D – 25%

8. Sum of all relative frequencies should be:

A) 0
B) 1
C) 10
D) 100

Answer: B – 1

9. Sum of all percentage frequencies should be:

A) 1%
B) 10%
C) 50%
D) 100%

Answer: D – 100%

10. The last less-than cumulative frequency is equal to:

A) First frequency
B) Class width
C) Total frequency
D) Class midpoint

Answer: C – Total Frequency

11. In exclusive classification 20–30, the observation 30 belongs to:

A) 20–30
B) 30–40
C) Both classes
D) Neither class

Answer: B – 30–40

12. The real limits of class 10–19, for data measured to the nearest unit, are:

A) 10–19
B) 9–20
C) 9.5–19.5
D) 10.5–18.5

Answer: C – 9.5–19.5

Numerical MCQs

13. Frequencies are 4, 6, 10, 8 and 2. What is N?

N = 4 + 6 + 10 + 8 + 2 = 30

Answer: 30

14. Less-than cumulative frequencies are 5, 12, 20, 31. What is the frequency of the third class?

Third frequency = 20 − 12 = 8

Answer: 8

15. If N = 250 and relative frequency of a class is 0.12, its frequency is:

f = 0.12 × 250 = 30

Answer: 30

16. If a class contains 45 observations out of 180, its percentage is:

45/180 × 100 = 25%

Answer: 25%

17. Class midpoint is 25 and class width is 10. For an equal-width continuous class, its limits are:

Half width = 10/2 = 5

Lower = 25 − 5 = 20
Upper = 25 + 5 = 30

Answer: 20–30

One-Minute Master Table

Term	Meaning / Formula
Raw Data	Data before processing
Frequency	Number of occurrences
Frequency Distribution	Classes/values with corresponding frequencies
Grouped Data	Data presented in frequency distribution form
Lower Class Limit	Lower end of class
Upper Class Limit	Upper end of class
Class Boundary	Real limit separating adjacent classes
Class Width	Upper Boundary − Lower Boundary
Class Mark	(Lower Limit + Upper Limit)/2
Relative Frequency	f/N
Percentage Frequency	(f/N) × 100
Cumulative Frequency	Successive total of frequencies
Less-than C.F.	Successive addition
More-than C.F.	Starts with N and decreases

చివరి నిమిషం పునశ్చరణ

Frequency → Number of occurrences

Frequency Distribution → Classes/values + Frequencies

Grouped Data → Frequency distribution form

Class Limits → Two ends of class

Class Width → Upper Boundary − Lower Boundary

Class Mark → (Lower + Upper)/2

Relative Frequency → f/N

Percentage → (f/N) × 100

Σf → N

Σ Relative Frequency → 1

Σ Percentage → 100%

Less-than C.F. → Add frequencies successively

Last Less-than C.F. → N

Frequency from C.F. → Successive difference

More-than C.F. → Starts with N and decreases

Inclusive → 10–19, 20–29

Exclusive → 10–20, 20–30

Golden Numerical Checks:

Σf = N
Σrf = 1
Σ% = 100
Last Less-than C.F. = N

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