STATISTICS NOTES

सांख्यिकी नोट्स

1. Measures of Central Tendency

केंद्रीय प्रवृत्ति के माप

Mean (Arithmetic Mean)

माध्य (अंकगणितीय माध्य)

The mean is the sum of all observations divided by the total number of observations.

माध्य सभी प्रेक्षणों के योग को कुल प्रेक्षणों की संख्या से भाग देकर प्राप्त किया जाता है।

Formula:

x̄ = (x₁ + x₂ + x₃ + ... + xₙ) / n

or

x̄ = Σxᵢ / n

Where:

x₁, x₂, x₃ ... = observations

n = total observations

Shortcut Method

x̄ = A + (Σdᵢ / n)

Where:

A = assumed mean

dᵢ = xᵢ − A

Mean of Frequency Distribution

x̄ = (Σfᵢxᵢ) / (Σfᵢ)

Where:

fᵢ = frequency

xᵢ = observation

Combined Mean

x̄ = (N₁X̄₁ + N₂X̄₂) / (N₁ + N₂)

Important Properties of Mean

माध्य के महत्वपूर्ण गुण

1. Sum of deviations from mean is zero.

Σ(xᵢ − x̄) = 0

1. Sum of squares of deviations is minimum.

Σ(xᵢ − x̄)² is minimum

1. Mean changes if every observation changes by a constant.
2. Mean is affected by extreme values.

Geometric Mean (G.M.)

ज्यामितीय माध्य

G.M. = (x₁ × x₂ × x₃ × ... × xₙ)^(1/n)

For frequency distribution:

G.M. = (x₁^f₁ × x₂^f₂ × ... × xₙ^fₙ)^(1/N)

Where:

N = f₁ + f₂ + ... + fₙ

Harmonic Mean (H.M.)

हरात्मक माध्य

H.M. = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

For frequency distribution:

H.M. = (f₁ + f₂ + ... + fₙ) / (f₁/x₁ + f₂/x₂ + ... + fₙ/xₙ)

Illustration 11.1

Question

If X = aU + bV, then find the mean of X.

Solution

X̄ = aŪ + bV̄

Answer: aŪ + bV̄

Illustration 11.2

Question

The mean of numbers 27 + x, 31 + x, 89 + x, 107 + x, 156 + x is 82. Find the mean of 130 + x, 126 + x, 68 + x, 50 + x, 1 + x.

Solution

82 = [(27 + x) + (31 + x) + (89 + x) + (107 + x) + (156 + x)] / 5

82 × 5 = 410 + 5x

410 = 410 + 5x

x = 0

Required mean:

= (130 + 126 + 68 + 50 + 1) / 5

= 375 / 5

= 75

Answer: 75

Illustration 11.3

Question

If arithmetic mean of x₁, x₂, x₃, ... xₙ is x̄, then find mean of ax₁ + b, ax₂ + b, ...

Solution

New Mean = ax̄ + b

Answer: ax̄ + b

Illustration 11.4

Question

Find weighted mean of first n natural numbers whose weights are squares of corresponding numbers.

Solution

Weighted Mean = (1³ + 2³ + ... + n³) / (1² + 2² + ... + n²)

Using formulas:

Σn² = n(n + 1)(2n + 1) / 6

Σn³ = [n(n + 1)/2]²

Final Answer:

3n(n + 1) / 2(2n + 1)

Illustration 11.5

Question

A student obtains 75%, 80%, and 85% in three subjects. Find minimum average after adding fourth subject.

Solution

Total marks = 75 + 80 + 85 = 240

Minimum average occurs when marks in fourth subject = 0.

Average = 240 / 4 = 60%

Answer: 60%

MEDIAN

मध्यिका

Median is the middle value when observations are arranged in ascending or descending order.

मध्यिका वह मध्य मान है जब आंकड़ों को आरोही या अवरोही क्रम में व्यवस्थित किया जाता है।

Median of Ungrouped Data

If n is odd:

Median = Value of ((n + 1)/2)th observation

If n is even:

Median = [Value of (n/2)th observation + Value of ((n/2) + 1)th observation] / 2

Median of Continuous Frequency Distribution

Median = l + [(N/2 − C)/f] × i

Where:

l = lower limit of median class

N = total frequency

C = cumulative frequency before median class

f = frequency of median class

i = class width

Illustration 11.8

Question

Mean weight of 7 students is 55 kg. Weights of 6 students are 52, 58, 55, 53, 56, 54. Find weight of 7th student.

Solution

Total weight = 55 × 7 = 385

Sum of six weights:

52 + 58 + 55 + 53 + 56 + 54 = 328

Weight of seventh student:

385 − 328 = 57

Answer: 57 kg

MODE

बहुलक

Mode is the observation having maximum frequency.

बहुलक वह मान है जिसकी आवृत्ति सबसे अधिक होती है।

Mode Formula

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × i

Where:

l = lower limit of modal class

f₁ = frequency of modal class

f₀ = frequency of preceding class

f₂ = frequency of succeeding class

i = class width

Relation Between Mean, Median and Mode

Mode = 3Median − 2Mean

or

Mean − Mode = 3(Mean − Median)

Illustration 11.11

Question

If Mode = 6 and Mean = 9, find Median.

Solution

Using formula:

6 = 3Median − 2 × 9

6 = 3Median − 18

3Median = 24

Median = 8

Answer: 8

Measures of Dispersion

प्रसरण के माप

The measures of dispersion are:

1. Range
2. Mean Deviation
3. Variance
4. Standard Deviation

Range

Range = Largest Value − Smallest Value

Mean Deviation

Ungrouped Data

Mean deviation about mean:

M.D.(x̄) = (1/n) Σ|xᵢ − x̄|

Mean deviation about median:

M.D.(M) = (1/n) Σ|xᵢ − M|

Illustration 11.12

Question

Find mean deviation about mean:

Size 20 21 22 23 24

Frequency 6 4 5 1 4

Solution

x̄ = 433 / 20 = 21.65

M.D. = 25 / 20 = 1.25

Answer: 1.25

VARIANCE

प्रसरण

Variance is the arithmetic mean of squares of deviations from mean.

प्रसरण माध्य से विचलनों के वर्गों का औसत है।

Formula:

σ² = (1/n) Σ(xᵢ − x̄)²

or

σ² = (1/n) Σxᵢ² − (Σxᵢ / n)²

STANDARD DEVIATION (S.D.)

मानक विचलन

Standard deviation is the positive square root of variance.

मानक विचलन प्रसरण का धनात्मक वर्गमूल होता है।

Formula:

σ = √[(1/n) Σ(xᵢ − x̄)²]

or

σ = √[(1/n) Σxᵢ² − (Σxᵢ / n)²]

Illustration 11.16

Question

Find standard deviation of data:

6, 5, 9, 13, 12, 8, 10

Solution

Σxᵢ = 63

Σxᵢ² = 619

N = 7

σ = √[(619/7) − (63/7)²]

= √[(619/7) − 81]

= √(52/7)

Answer: √(52/7)

Important Short Notes

महत्वपूर्ण शॉर्ट नोट्स

1. Mean is affected by extreme values.
2. माध्य चरम मानों से प्रभावित होता है।
3. Median is positional average.
4. मध्यिका स्थानिक औसत है।
5. Mode is the most frequent observation.
6. बहुलक सर्वाधिक आवृत्ति वाला मान है।
7. In symmetric distribution:

Mean = Median = Mode

1. In positively skewed distribution:

Mean > Median > Mode

1. In negatively skewed distribution:

Mean < Median < Mode

Statistics Exercise MCQs for SSC & Railway Exams

सांख्यिकी अभ्यास MCQs SSC और Railway Exams के लिए

1. Question

Coefficients of variation of 2 distributions are 50 and 60, and their arithmetic means are 30 and 25, respectively. Difference of their standard deviations is:

2 वितरणों के variation coefficients क्रमशः 50 और 60 हैं, तथा उनके arithmetic mean क्रमशः 30 और 25 हैं। उनके standard deviations का अंतर क्या होगा?

Options:

(1) 0

(2) 1

(3) 1.5

(4) 2.5

Answer: (1) 0

2. Question

The mean of a set of numbers is X̄. If each number is divided by 3, then the new mean is:

किसी संख्याओं के समूह का mean X̄ है। यदि प्रत्येक संख्या को 3 से भाग दिया जाए, तो नया mean क्या होगा?

Options:

(1) X̄

(2) X̄ + 3

(3) 3X̄

(4) X̄ / 3

Answer: (4) X̄ / 3

3. Question

The AM of the series 1, 2, 4, 8, 16, ..., 2ⁿ is:

श्रृंखला 1, 2, 4, 8, 16, ..., 2ⁿ का AM क्या होगा?

Options:

(1) (2ⁿ − 1) / n

(2) (2ⁿ⁺¹ − 1) / (n + 1)

(3) (2ⁿ + 1) / n

(4) (2ⁿ − 1) / (n + 1)

Answer: (2) (2ⁿ⁺¹ − 1) / (n + 1)

4. Question

The mean of n items is X̄. If the first item is increased by 1, second by 2 and so on, then the new mean is:

n वस्तुओं का mean X̄ है। यदि पहली वस्तु में 1, दूसरी में 2 और इसी प्रकार आगे वृद्धि की जाए, तो नया mean क्या होगा?

Options:

(1) X̄ + n

(2) X̄ + n/2

(3) X̄ + (n + 1)/2

(4) None of these

Answer: (3) X̄ + (n + 1)/2

5. Question

In a moderately skewed distribution, the values of mean and median are 5 and 6, respectively. The value of mode in such a situation is approximately equal to:

एक moderately skewed distribution में mean और median क्रमशः 5 और 6 हैं। ऐसी स्थिति में mode का मान लगभग कितना होगा?

Options:

(1) 8

(2) 11

(3) 6

(4) None of these

Answer: (1) 8

6. Question

For a normal distribution, if the mean is M, mode is M₀ and median is Mᵈ, then:

एक normal distribution में यदि mean M, mode M₀ और median Mᵈ है, तो:

Options:

(1) M > Mᵈ > M₀

(2) M < Mᵈ < M₀

(3) M = MᵈM₀

(4) M = Mᵈ = M₀

Answer: (4) M = Mᵈ = M₀

7. Question

The following data give the distribution of heights of students. The median of the distribution is:

नीचे दिया गया डेटा छात्रों की height distribution को दर्शाता है। इस distribution की median क्या होगी?

Height in cm: 160, 150, 152, 161, 156, 154, 155

Number of students: 12, 8, 4, 4, 3, 3, 7

Options:

(1) 154

(2) 155

(3) 160

(4) 161

Answer: (2) 155

8. Question

An automobile driver travels from a plain to a hill station 120 km away at an average speed of 30 km per hour. Then he makes the return trip at an average speed of 25 km per hour. He covers another 120 km on the plain at an average speed of 50 km per hour. His average speed over the entire distance of 360 km will be:

एक automobile driver plain से 120 km दूर hill station तक 30 km/h की average speed से जाता है। फिर वह 25 km/h की average speed से वापस आता है। इसके बाद वह plain पर 120 km की दूरी 50 km/h की average speed से तय करता है। कुल 360 km दूरी के लिए उसकी average speed क्या होगी?

Options:

(1) (30 + 25 + 50) / 3

(2) (1/30 + 1/25 + 1/50) / 3

(3) 3 / (1/30 + 1/25 + 1/50)

(4) None of these

Answer: (3) 3 / (1/30 + 1/25 + 1/50)

9. Question

The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is:

डेटा 3, 10, 10, 4, 7, 10, 5 का mean से mean deviation क्या है?

Options:

(1) 2

(2) 2.57

(3) 3

(4) 3.75

Answer: (2) 2.57

10. Question

When tested, the lives in hours of 5 bulbs were noted as follows: 1357, 1090, 1666, 1494, 1623. The mean deviation in hours from their mean is:

जांच करने पर 5 bulbs की life hours में इस प्रकार मिली: 1357, 1090, 1666, 1494, 1623. इनके mean से mean deviation क्या होगा?

Options:

(1) 178

(2) 179

(3) 220

(4) 356

Answer: (1) 178

11. Question

Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 53, 39, 40, 65, 59. The mean deviation from the median is:

एक mathematics test में 9 students द्वारा प्राप्त marks हैं: 50, 69, 20, 33, 53, 39, 40, 65, 59. Median से mean deviation क्या होगा?

Options:

(1) 9

(2) 10.5

(3) 12.67

(4) 14.76

Answer: (3) 12.67

12. Question

If the mean of the distribution is 2.6, then the value of y is:

यदि distribution का mean 2.6 है, तो y का मान क्या होगा?

Variate x: 1, 2, 3, 4, 5

Frequency of x: 4, 5, y, 1, 2

Options:

(1) 24

(2) 13

(3) 8

(4) 3

Answer: (3) 8

13. Question

If the mean of the set of numbers x₁, x₂, x₃, ..., xₙ is x̄, then the mean of the numbers xᵢ + 2i, 1 ≤ i ≤ n is:

यदि संख्याओं x₁, x₂, x₃, ..., xₙ का mean x̄ है, तो संख्याओं xᵢ + 2i, 1 ≤ i ≤ n का mean क्या होगा?

Options:

(1) x̄ + 2n

(2) x̄ + n + 1

(3) x̄ + 2

(4) x̄ + n

Answer: (2) x̄ + n + 1

14. Question

The harmonic mean of 4, 8, 16 is:

4, 8, 16 का harmonic mean क्या होगा?

Options:

(1) 6.4

(2) 6.7

(3) 6.85

(4) 7.8

Answer: (3) 6.85

15. Question

The average of n numbers x₁, x₂, x₃, ..., xₙ is M. If xₙ is replaced by x′, then new average is:

n संख्याओं x₁, x₂, x₃, ..., xₙ का average M है। यदि xₙ को x′ से बदल दिया जाए, तो नया average क्या होगा?

Options:

(1) M − xₙ + x′

(2) (nM − xₙ + x′) / n

(3) [(n − 1)M + x′] / n

(4) (M − xₙ + x′) / n

Answer: (2) (nM − xₙ + x′) / n

16. Question

The following data gives the distribution of height of students. The median of the distribution is:

नीचे दिया गया डेटा students की height distribution को दर्शाता है। इस distribution की median क्या होगी?

Height in cm: 160, 150, 152, 161, 156, 154, 155

Number of students: 12, 8, 4, 4, 3, 3, 7

Options:

(1) 154

(2) 155

(3) 160

(4) 161

Answer: (2) 155

17. Question

For a slightly asymmetric distribution, mean and median are 5 and 6, respectively. What is its mode?

एक slightly asymmetric distribution में mean और median क्रमशः 5 और 6 हैं। इसका mode क्या होगा?

Options:

(1) 5

(2) 6

(3) 7

(4) 8

Answer: (4) 8

18. Question

Runs scored by a batsman in 10 innings are: 38, 70, 48, 34, 42, 55, 63, 46, 54, 44. The mean deviation is:

एक बल्लेबाज द्वारा 10 innings में बनाए गए runs हैं: 38, 70, 48, 34, 42, 55, 63, 46, 54, 44. Mean deviation क्या है?

Options:

(1) 8.6

(2) 6.4

(3) 10.6

(4) 9.6

Answer: (1) 8.6

19. Question

If μ is the mean of distribution (v₁, ρ₁), then Σf(vᵢ − μ) =

यदि μ वितरण (v₁, ρ₁) का mean है, तो Σf(vᵢ − μ) = ?

Options:

(1) M.D.

(2) S.D.

(3) 0

(4) Relative frequency

Answer: (3) 0

20. Question

The range of the following set of observations 2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3 is:

प्रेक्षणों 2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3 का range क्या है?

Options:

(1) 11

(2) 7

(3) 5.5

(4) 6

Answer: (2) 7

21. Question

If each observation of a raw data whose variance is σ² is multiplied by h, then the variance of the new set is:

यदि किसी raw data का variance σ² है और प्रत्येक observation को h से गुणा किया जाए, तो नए set का variance क्या होगा?

Options:

(1) σ²

(2) h²σ²

(3) hσ²

(4) h + σ²

Answer: (2) h²σ²

22. Question

If a variable x takes values 0, 1, 2, ..., n with frequencies proportional to the binomial coefficients nC₀, nC₁, nC₂, ..., nCₙ, then var(X) is:

यदि variable x के मान 0, 1, 2, ..., n हों तथा उनकी frequencies nC₀, nC₁, nC₂, ..., nCₙ के proportional हों, तो var(X) क्या होगा?

Options:

(1) (n² − 1)/12

(2) n/2

(3) n/4

(4) None of these

Answer: (3) n/4

23. Question

The variance of the data 2, 4, 6, 8, 10 is:

डेटा 2, 4, 6, 8, 10 का variance क्या है?

Options:

(1) 6

(2) 7

(3) 8

(4) None of these

Answer: (3) 8

24. Question

If the standard deviation of 0, 1, 2, 3, ..., 9 is K, then the standard deviation of 10, 11, 12, ..., 19 is:

यदि 0, 1, 2, 3, ..., 9 का standard deviation K है, तो 10, 11, 12, ..., 19 का standard deviation क्या होगा?

Options:

(1) K

(2) K + 10

(3) K + √10

(4) 10K

Answer: (1) K

25. Question

For a given distribution of marks, the mean is 35.16 and standard deviation is 19.76. The coefficient of variation is:

एक marks distribution में mean 35.16 तथा standard deviation 19.76 है। Coefficient of variation क्या होगा?

Options:

(1) 35.16 / 19.76

(2) 19.76 / 35.16

(3) (35.16 / 19.76) × 100

(4) (19.76 / 35.16) × 100

Answer: (4) (19.76 / 35.16) × 100

26. Question

The mean and the SD of 1, 2, 3, 4, 5, 6 are:

1, 2, 3, 4, 5, 6 का mean और SD क्या है?

Options:

(1) 7/2 , √(35/12)

(2) 3 , 3

(3) 7/2 , √3

(4) 3 , 35/12

Answer: (1) 7/2 , √(35/12)

27. Question

The standard deviation of 25 numbers is 40. If each number is increased by 5, then the new standard deviation will be:

25 numbers का standard deviation 40 है। यदि प्रत्येक संख्या में 5 जोड़ दिया जाए, तो नया standard deviation क्या होगा?

Options:

(1) 40

(2) 45

(3) 40 + 21/25

(4) None of these

Answer: (1) 40

28. Question

Consider any set of observations x₁, x₂, x₃, ..., x₁₀₁. It is given that x₁ < x₂ < x₃ < ... < x₁₀₀ < x₁₀₁. Then the mean deviation of this set about a point k is minimum when k equals:

किसी observations set x₁, x₂, x₃, ..., x₁₀₁ में x₁ < x₂ < x₃ < ... < x₁₀₀ < x₁₀₁ है। तब इस set का mean deviation point k के बारे में minimum होगा जब k बराबर होगा:

Options:

(1) x₁

(2) x₅₁

(3) (x₁ + x₂ + ... + x₁₀₁)/101

(4) x₅₀

Answer: (2) x₅₁

29. Question

For (2n + 1) observations x₁, −x₁, x₂, −x₂, ..., xₙ, −xₙ and 0, where all x’s are distinct, let SD and MD denote standard deviation and median respectively. Then which is always true?

(2n + 1) observations x₁, −x₁, x₂, −x₂, ..., xₙ, −xₙ और 0 में, जहाँ सभी x distinct हैं, SD तथा MD क्रमशः standard deviation और median हैं। निम्न में से क्या सदैव सत्य है?

Options:

(1) SD < MD

(2) SD > MD

(3) SD = MD

(4) Nothing can be said

Answer: (2) SD > MD

30. Question

Let r be the range and S² = [1/(n − 1)] Σ(xᵢ − x̄)² be the SD of observations x₁, x₂, ..., xₙ. Then:

यदि r range है तथा S² = [1/(n − 1)] Σ(xᵢ − x̄)² observations x₁, x₂, ..., xₙ का SD है, तो:

Options:

(1) S ≤ r√(n/(n − 1))

(2) S = r√(n/(n − 1))

(3) S ≥ r√(n/(n − 1))

(4) None of these

Answer: (1) S ≤ r√(n/(n − 1))

31. Question

The SD of a variate x is σ. The SD of variate (ax + b)/c, where a, b, c are constants, is:

यदि variate x का SD σ है, तो variate (ax + b)/c का SD क्या होगा?

Options:

(1) (a/c)σ

(2) |a/c|σ

(3) (a²/c²)σ

(4) None of these

Answer: (2) |a/c|σ

32. Question

The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is:

डेटा 6, 5, 9, 13, 12, 8, 10 का standard deviation क्या है?

Options:

(1) √(52/7)

(2) 52/7

(3) √6

(4) 6

Answer: (1) √(52/7)

33. Question

The mean of 100 observations is 50 and their standard deviation is 5. The sum of squares of all observations is:

100 observations का mean 50 तथा standard deviation 5 है। सभी observations के squares का योग क्या होगा?

Options:

(1) 50000

(2) 250000

(3) 252500

(4) 255000

Answer: (3) 252500

34. Question

Standard deviation for first 10 natural numbers is:

पहली 10 natural numbers का standard deviation क्या है?

Options:

(1) 5.5

(2) 3.87

(3) 2.97

(4) 2.87

Answer: (4) 2.87

35. Question

Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. If 1 is added to each number, the variance of the numbers obtained is:

संख्याएँ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 लें। यदि प्रत्येक संख्या में 1 जोड़ दिया जाए, तो प्राप्त संख्याओं का variance क्या होगा?

Options:

(1) 6.5

(2) 2.87

(3) 3.87

(4) 8.25

Answer: (4) 8.25

36. Question

Consider the first 10 positive integers. If we multiply each number by −1 and then add 1 to each number, the variance of the obtained numbers is:

पहली 10 positive integers लें। यदि प्रत्येक संख्या को −1 से गुणा कर उसमें 1 जोड़ दिया जाए, तो प्राप्त संख्याओं का variance क्या होगा?

Options:

(1) 8.25

(2) 6.5

(3) 3.87

(4) 2.87

Answer: (1) 8.25

37. Question

The following information relates to a sample of size 60:

Σx² = 18000, Σx = 960

The variance is:

निम्न जानकारी 60 size के sample से संबंधित है:

Σx² = 18000, Σx = 960

Variance क्या होगा?

Options:

(1) 6.63

(2) 16

(3) 22

(4) 44

Answer: (4) 44

38. Question

The standard deviation of some temperature data in °C is 5. If the data were converted into °F, the new variance would be:

किसी temperature data का standard deviation °C में 5 है। यदि data को °F में बदला जाए, तो नया variance क्या होगा?

Options:

(1) 81

(2) 57

(3) 36

(4) 25

Answer: (1) 81

39. Question

What is the standard deviation of the following data?

निम्न डेटा का standard deviation क्या होगा?

Measurement: 0–10, 10–20, 20–30, 30–40

Frequency: 1, 3, 4, 2

Options:

(1) 81

(2) 7.6

(3) 8.1

(4) 2.26

Answer: (3) 8.1