Five workers in a shop earn ₹300, ₹300, ₹400, ₹400, and ₹20,000 per day (the owner). The owner claims the 'average' worker earns ₹4,280 and they have nothing to complain about. Which statistical measure proves the owner is misleading them?

Answer: The Median. The Mean is easily skewed by extreme outliers (20000). The Median represents the actual 'middle' worker. Sorted: 300, 300, 400, 400, 20000. Median is 400. This reflects true central tendency here.

A shoe store wants to stock up its inventory. Finding which of the following measures for the 'Shoe Size' data of past customers will be purely the most useful for business restock?

Answer: Mode. Mode indicates the highest frequency (most popular size). A shoe store needs to order the most popular size, not the 'average' size (a 7.84 size shoe doesn't exist).

A factory machine produces items. The lengths of 5 items produced randomly are: 5.1cm, 5.0cm, 5.2cm, 5.1cm, 9.5cm. To measure how 'inconsistent' the machine is, the engineer computes the Range. What is it?

Answer: 4.5 cm. Range = Highest - Lowest = 9.5 - 5.0 = 4.5 cm.

A student took 6 tests. Her scores were exactly exactly the same in every test (75). What is the Standard Deviation or Variance of her scores?

Answer: 0. Standard deviation measures distance from the mean. Since all scores are 75, the mean is 75, and no score deviates from it. Hence, Zero dispersion.

A cab driver records daily earnings: ₹500, ₹450, ₹600, ₹400, ₹550. What is his Median earning?

Answer: 500. First, SORT: 400, 450, 500, 550, 600. Middle number is 500.

A cricket team's average score in 5 matches is 250. If the score of the 6th match is exactly 250, what happens to the overall average?

Answer: Remains the same. Adding an element that is exactly equal to the existing Mean does not change the Mean at all.

In an exam, a teacher gave everyone 5 bonus marks. What happens to the Range of the marks?

Answer: It remains totally unchanged. Range = Highest - Lowest. If Highest becomes H+5 and Lowest becomes L+5. Range = (H+5) - (L+5) = H - L. It stays exactly the same.

A class tracks weather. Temperatures: 30, 31, 30, 32, 33, 30. Which is true?

Answer: Mean > Mode. Sorted: 30, 30, 30, 31, 32, 33. Mode is 30. Sum = 186/6 = 31. Mean is 31. Thus, Mean (31) > Mode (30).

The average age of a family of 4 is 25. A newborn baby arrives. What is the immediate new average age of the family?

Answer: 20. Total age of 4 = 100. Newborn is 0. New Total = 100. New Members = 5. New Average = 100/5 = 20.

A data analyst notices a dataset has TWO numbers occurring 10 times each (the highest frequency). This distribution is called:

Answer: Bimodal. When a dataset has two modes, it is called Bimodal.

Find the median of: 15, 22, 10, 8, 30, 42.

Answer: 18.5. Sorted: 8, 10, 15, 22, 30, 42. N=6 (Even!). Middle two are 15 and 22. Average = (15+22)/2 = 37/2 = 18.5.

Mean of 10 numbers is 15. If one number 35 is entirely completely removed, find the new mean.

Answer: 12.77. Total = 150. Remove 35 → 115. New count = 9. 115 / 9 = 12.77.

Using empirical relation, find the Mode if Mean = 50 and Median = 40.

Answer: 20. Mode = 3(Median) - 2(Mean) → 3(40) - 2(50) = 120 - 100 = 20.

The mean of 5 observations x, x+2, x+4, x+6, x+8 is 11. Find the value of x.

Answer: 7. This is an arithmetic progression. The mean is simply the middle term (x+4). x+4 = 11 → x = 7. Or algebraic: (5x + 20)/5 = 11 → x+4=11 → x=7.

Find the mean of the first 9 prime numbers.

Answer: 11 1/9. Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23. Sum = 100. Mean = 100/9 = 11 1/9.

A class of 20 boys scores an average of 60%. A class of 30 girls scores an average of 70%. What is the combined total class average?

Answer: 66%. Weighted Mean = [(20 × 60) + (30 × 70)] / (20+30) = (1200 + 2100) / 50 = 3300 / 50 = 66.

The mean of the data set is 8. The individual values are: 4, 7, 7, 8, x, 11, 12, 10. Find x.

Answer: 5. Total values = 8. Target Sum = 8 × 8 = 64. Current Sum = 59. x = 64 - 59 = 5.

If every single observation in a data set is strictly multiplied by 3, the new mean will be:

Answer: Multiplied by 3. Mean scales linearly with multiplication and division. If elements are multiplied by K, Mean is multiplied by K.

Find the mean of the squares of the first 5 natural numbers.

Answer: 11. Squares: 1, 4, 9, 16, 25. Sum = 55. Mean = 55 / 5 = 11. Or formula: n(n+1)(2n+1)/6.

Which frequency graph uses class boundaries instead of class limits to ensure no gaps between bars?

Answer: Histogram. Histograms represent continuous data and thus have no spaces between vertical bars.

The mean of 50 observations was 36. It was found later that an observation 48 was wrongly taken as 23. What is the correct Mean?

Answer: 36.5. Difference to correct = +25. This +25 is distributed over 50 items. Mean increases by 25/50 = 0.5. Old mean 36 + 0.5 = 36.5.

What is the Median of the first 10 whole numbers?

Answer: 4.5. Whole numbers start at ZERO. 0,1,2,3,4,5,6,7,8,9. N=10. Middle two are 4 and 5. Average = 4.5.

A dataset has Range 45. The highest value is 82. What is the lowest value?

Answer: 37. Range = High - Low → 45 = 82 - Low → Low = 82 - 45 = 37.

If the algebraic sum of the deviations from 'X' of N numbers is exactly ZERO, then X is necessarily the:

Answer: Mean. The sum of deviations explicitly from the Arithmetic Mean is mathematically always Zero. It perfectly balances positive and negative distances.

The median of a set of 11 distinct numbers is 30. If every number greater than 30 is suddenly multiplied by 5, what is the new Median?

Answer: 30. The numbers greater than 30 stay greater than 30. They move further right, but the physical 6th position (the median) does not shift. It remains 30.

Mean of a dataset is M. If each observation x<sub>i</sub> is replaced by ax<sub>i</sub> + b, where a and b are constants, the new mean is:

Answer: aM + b. Linear transformation exactly applies to the mean. The new mean literally becomes aM + b.

In a class, the ratio of boys to girls is 3:2. Average marks of boys is A, and girls is A+5. If class average is 72, find the average of boys (A).

Answer: 70. Weighted Mean = [3A + 2(A+5)] / 5 = 72 → 5A + 10 = 360 → 5A = 350 → A = 70.

The median of observations 11, 12, 14, 18, x+2, x+4, 30, 32, 35, 41 arranged in ALL ascending order is 24. Find the Mode.

Answer: There is no Mode. N=10. Median = (x+2 + x+4)/2 = x+3. x+3 = 24 → x=21. The sequence is 11,12,14,18,23,25,30,32,35,41. Every number appears once. There is no mode.

A bat and ball together cost ₹1100. The bat costs exactly ₹1000 more than the ball. If this is treated as a 2-item dataset, what is the Range?

Answer: 1000. Ball = 50. Bat = 1050. Total = 1100. Range = 1050 - 50 = 1000. Alternatively, range IS the difference, which is already given as 1000.

Calculate the arithmetic mean of the geometric mode and median of the subset: {2, 2, 8, 8, 8, 2, 2, 8}.

Answer: 5. Sorted: 2, 2, 2, 2, 8, 8, 8, 8. Both 2 and 8 appear 4 times. It is Bimodal (Modes = 2 and 8). The question states 'geometric mode', meaning no unique mode. But wait! Let's find Median: middle is (2+8)/2 = 5. If we average Modal concepts or perhaps the question is a trick. Let's look at modes (2,8) and median (5). Average of them could be complex. Let's fix the question logical flaw and just say 'Median'. Median = (2+8)/2 = 5.

Home/Rajasthan Librarian Exam/Quantitative Aptitude/

Statistics & Data Analysis

Master the art of extracting meaning from numbers. Dive deep into Mean, Median, Mode, Range, and advanced Data Interpretation.

Expert Answer & Key Takeaways

Master the art of extracting meaning from numbers. Dive deep into Mean, Median, Mode, Range, and advanced Data Interpretation.

1. Measures of Central Tendency

Methods to find the 'center' of a dataset.

Mean (Average): The mathematical center. Sum of all values divided by total count. Highly sensitive to extreme values (outliers).
Median: The physical middle number when data is sorted. Best used when data has extreme outliers.
Mode: The most frequently occurring number. Used for finding popularity.

Example:

Q: Find the Mean, Median, and Mode of: 2, 2, 5, 10, 11.

Solution: Mean = 30/5 = 6. Median (sorted middle) = 5. Mode (most frequent) = 2.

2. The Empirical Relationship

For a moderately skewed distribution, the three measures of central tendency are connected by an approximate formula:

Mode = 3(Median) - 2(Mean)

Example:

Q: If the mean is 20 and median is 22, what is the approximate mode?

Solution: Mode = 3(22) - 2(20) = 66 - 40 = 26.

3. Measures of Dispersion (Range)

While central tendency tells us the 'middle', dispersion tells us the 'spread'.

Range: Highest Value − Lowest Value. A high range means volatile data; a low range means consistent data.

Example:

Q: A batsman scores 12, 18, 50, 100, 2 in five matches. Find the range.

Solution: Range = Highest (100) - Lowest (2) = 98.

4. Combined / Weighted Mean

When combining two groups with different sizes and means.

Combined Mean = N 1 A 1 + N 2 A 2 N 1 + N 2

Example:

Q: Class A (20 students) averaged 50. Class B (30 students) averaged 40. Find overall average.

Solution: (20 × 50 + 30 × 40) / (20 + 30) = (1000 + 1200) / 50 = 2200 / 50 = 44.

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Statistics & Data Analysis

Expert Answer & Key Takeaways

1. Measures of Central Tendency

2. The Empirical Relationship

3. Measures of Dispersion (Range)

4. Combined / Weighted Mean

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