PRC-2 · Chapter 8
Statistical Measures of Data MCQs with Answers
60 multiple-choice questions on Statistical Measures of Data for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.
Practise this chapter interactivelyQuestion 1
If an instructor decides to arbitrarily increase every student's test score by exactly 5 marks, what will happen to the arithmetic mean of the test scores?
- A) It will remain completely unchanged.
- B) It will increase by exactly 5 units.
- C) It will increase by 25 units.
- D) It cannot be determined without the original scores.
Show answer & explanation
Answer: B) It will increase by exactly 5 units.
A key mathematical property of the arithmetic mean is that it is affected by a change of origin. If a constant 'c' is added to every observation in a dataset, the mean itself simply increases by that exact constant.
Question 2
Find the median of the following ungrouped data set: 14, 8, 22, 19, 5.
- A) 13.6
- B) 14
- C) 22
- D) 8
Show answer & explanation
Answer: B) 14
To find the median, first sort the data in ascending order: 5, 8, 14, 19, 22. Because there is an odd number of observations (5), the median is exactly the middle value, which is 14.
Question 3
In a highly skewed distribution, which statistical measure of central tendency is typically considered the most robust and representative?
- A) Arithmetic Mean
- B) Geometric Mean
- C) Median
- D) Mode
Show answer & explanation
Answer: C) Median
The median is largely unaffected by extreme outliers or heavily skewed data, making it the most representative measure of the 'center' compared to the mean, which is heavily distorted by extreme values.
Question 4
Group X consists of 40 items with a mean of 50. Group Y consists of 60 items with a mean of 70. What is the combined mean of the two groups?
- A) 60
- B) 62
- C) 64
- D) 65
Show answer & explanation
Answer: B) 62
Using the combined mean formula: ((n1 * Mean1) + (n2 * Mean2)) / (n1 + n2). This gives ((40 * 50) + (60 * 70)) / 100 = (2000 + 4200) / 100 = 6200 / 100 = 62.
Question 5
If the calculated variance for a specific set of financial data is strictly 144, what is the standard deviation of that data?
- A) 12
- B) -12
- C) 144
- D) 20,736
Show answer & explanation
Answer: A) 12
Standard deviation is mathematically defined as the positive square root of the variance. The square root of 144 is 12. Standard deviation cannot be a negative number.
Question 6
What is the mathematical sum of the deviations of all individual observations from their arithmetic mean?
- A) 1
- B) The variance
- C) Exactly zero
- D) The standard deviation
Show answer & explanation
Answer: C) Exactly zero
A fundamental property of the arithmetic mean is that the sum of the differences (deviations) between each data point and the mean is always precisely zero.
Question 7
Which specific measure of central tendency is most mathematically appropriate for averaging rates of change, speed, or unit prices?
- A) Geometric Mean
- B) Harmonic Mean
- C) Arithmetic Mean
- D) Median
Show answer & explanation
Answer: B) Harmonic Mean
The harmonic mean is specifically preferred for dealing with ratios, rates of change, and speeds (e.g., calculating average travel speed over different distances).
Question 8
Which of the following values is perfectly equivalent to the Second Quartile (Q2) of a dataset?
- A) The Mean
- B) The Mode
- C) The Median
- D) The 25th Percentile
Show answer & explanation
Answer: C) The Median
The second quartile (Q2) splits the data exactly in half (the 50th percentile), making it mathematically and conceptually identical to the median.
Question 9
How is the Interquartile Range (IQR) formally calculated?
- A) Maximum Value minus Minimum Value
- B) Third Quartile (Q3) minus First Quartile (Q1)
- C) (Third Quartile minus First Quartile) / 2
- D) Mean minus Median
Show answer & explanation
Answer: B) Third Quartile (Q3) minus First Quartile (Q1)
The Interquartile Range measures the spread of the middle 50% of the dataset. It is strictly calculated by subtracting the lower quartile (Q1) from the upper quartile (Q3).
Question 10
A dataset contains the values 10, 10, 10, 10, and 10. What is the standard deviation of this set?
- A) 10
- B) 5
- C) 0
- D) 1
Show answer & explanation
Answer: C) 0
Standard deviation measures the dispersion or spread of data around the mean. Since all values are identical, there is absolutely no variation, meaning the standard deviation is 0.
Question 11
Which of the following statements regarding the statistical measure of 'variance' is universally true?
- A) Variance can be a negative number.
- B) Variance is always positive or zero.
- C) Variance is measured in the exact same units as the original data.
- D) Variance is completely unaffected by multiplication.
Show answer & explanation
Answer: B) Variance is always positive or zero.
Because variance is calculated by squaring the deviations from the mean, the result can never be negative. It is always a positive number or exactly zero (if there is no spread).
Question 12
If you multiply every observation in a dataset by a constant 'k', what will happen to the new standard deviation?
- A) It remains unchanged.
- B) It is multiplied by k.
- C) It is multiplied by k squared.
- D) It increases by k.
Show answer & explanation
Answer: B) It is multiplied by k.
Standard deviation is affected by a change of scale. If every value is multiplied by a constant 'k', the standard deviation of the new set is exactly the old standard deviation multiplied by 'k'.
Question 13
Which measure of central tendency is defined as the value that appears most frequently in a given dataset?
- A) Mean
- B) Median
- C) Mode
- D) Range
Show answer & explanation
Answer: C) Mode
The mode is strictly defined as the observation that occurs with the highest frequency in a dataset. A dataset can have one mode, multiple modes, or no mode at all.
Question 14
For a symmetrical (normal) distribution curve, which relationship between the mean, median, and mode is correct?
- A) Mean > Median > Mode
- B) Mean < Median < Mode
- C) Mean = Median = Mode
- D) Mode is zero
Show answer & explanation
Answer: C) Mean = Median = Mode
In a perfectly symmetrical, normal distribution, the data is evenly balanced around the center, causing the mean, median, and mode to all fall on the exact same numerical value.
Question 15
The mean of 5 observations is incorrectly recorded as 40. Upon review, it was found that a value of 25 was mistakenly read as 50. What is the correct mean?
- A) 35
- B) 45
- C) 30
- D) 38
Show answer & explanation
Answer: A) 35
Incorrect total sum = 5 * 40 = 200. The error added 25 too much (50 - 25). The correct sum is 200 - 25 = 175. The correct mean is 175 / 5 = 35.
Question 16
If a researcher multiplies every single observation in a dataset by exactly 5, what will mathematically happen to the arithmetic mean of that new dataset?
- A) It will increase by 5.
- B) It will increase by 5 times.
- C) It will remain completely unaltered.
- D) It will decrease by 5 times.
Show answer & explanation
Answer: B) It will increase by 5 times.
A fundamental property of the arithmetic mean is that if each observation is multiplied by a constant, the new mean increases by that exact multiple.
Question 17
If every observation in a statistical sample is divided by 10, how is the new arithmetic mean affected?
- A) It is mathematically multiplied by 10.
- B) It remains completely unchanged.
- C) It is divided by 10.
- D) It increases by exactly 10.
Show answer & explanation
Answer: C) It is divided by 10.
If each observation in a dataset is divided by a specific constant like 10, the resulting arithmetic mean of the new set will also be divided by 10.
Question 18
Which of the following statements accurately describes a skewed distribution curve?
- A) Frequencies run up to a maximum at one end of the range.
- B) It departs from symmetry and frequencies tend to pile up at one or the other end of the curve.
- C) Values are at equal distance from a central maximum.
- D) It has maximum frequencies at both extreme ends.
Show answer & explanation
Answer: B) It departs from symmetry and frequencies tend to pile up at one or the other end of the curve.
A skewed curve departs from perfect symmetry, causing the data frequencies to heavily pile up at one extreme end of the range.
Question 19
How is a perfectly symmetrical frequency distribution defined visually?
- A) It departs heavily from symmetry.
- B) Frequencies pile up exclusively at the right end.
- C) It represents values that are at equal distance from a central maximum value.
- D) Frequencies run up to a maximum at one end.
Show answer & explanation
Answer: C) It represents values that are at equal distance from a central maximum value.
In a symmetrical distribution, the data points represent values that are plotted at equal distance mirroring a central maximum value.
Question 20
Which of the following best defines the 'standard error' in statistical analysis?
- A) The variance of a single sample.
- B) The standard deviation of the sampling distribution of the means.
- C) The median of a population.
- D) The mode of a grouped distribution.
Show answer & explanation
Answer: B) The standard deviation of the sampling distribution of the means.
The standard deviation of the sampling distribution of the means is specifically and technically called the standard error.
Question 21
If a constant value of 3 is systematically subtracted from every observation of variable Y, what will happen to the calculated correlation coefficient (r)?
- A) It will increase by 3.
- B) It will decrease by 3.
- C) It will remain completely unchanged.
- D) It will multiply by 3.
Show answer & explanation
Answer: C) It will remain completely unchanged.
The correlation coefficient (r) is independent of the change of origin, meaning adding or subtracting constants from the observations leaves it completely unchanged.
Question 22
Which of the following describes a U-shaped frequency distribution curve?
- A) Frequencies pile up heavily at one end.
- B) It has maximum frequencies occurring at both ends of the range and a minimum frequency towards the center.
- C) It is perfectly symmetrical with a single central peak.
- D) It represents equal frequencies across all possible classes.
Show answer & explanation
Answer: B) It has maximum frequencies occurring at both ends of the range and a minimum frequency towards the center.
A U-shaped curve is strictly characterized by having maximum frequencies at the extreme ends of the range and minimum frequencies dipping in the center.
Question 23
When is the arithmetic mean generally considered the most appropriate statistical measure?
- A) When data is heavily skewed.
- B) When the data contains massive extreme outliers.
- C) As the best single figure to describe symmetrical data.
- D) When analyzing purely qualitative survey data.
Show answer & explanation
Answer: C) As the best single figure to describe symmetrical data.
The arithmetic mean is widely considered the best single mathematical figure to describe a typical, non-skewed dataset.
Question 24
What is the true mathematical relationship between the variance and the standard deviation?
- A) Variance is the square root of the standard deviation.
- B) Standard deviation is the square root of the variance.
- C) They are always identical numbers.
- D) Variance is always exactly double the standard deviation.
Show answer & explanation
Answer: B) Standard deviation is the square root of the variance.
The variance can never be smaller than the standard deviation (except when the value is a fraction between 0 and 1), because the standard deviation is defined as the mathematical square root of the variance.
Question 25
A J-shaped statistical curve is essentially characterized by which of the following?
- A) It has maximum frequencies occurring at both ends of the range.
- B) It represents values at equal distance from a central maximum.
- C) It has frequencies that run up to a maximum at one end of the range.
- D) It depicts a completely constant frequency.
Show answer & explanation
Answer: C) It has frequencies that run up to a maximum at one end of the range.
A J-shaped frequency curve mathematically displays frequencies that consistently run up to a single maximum peak at one extreme end of the data range.
Question 26
If the variance of a dataset is mathematically calculated to be exactly zero, what does this imply about the dataset?
- A) All observations are completely identical with no dispersion.
- B) The arithmetic mean is also zero.
- C) The standard deviation is negative.
- D) The dataset has an infinite endless range.
Show answer & explanation
Answer: A) All observations are completely identical with no dispersion.
While it is extremely rare, variance can mathematically be zero if and only if there is absolutely no deviation from the mean, meaning all data points are identical.
Question 27
If you multiply each observation of variable X by 100, what will happen to a correlation coefficient of 0.9?
- A) It will increase to 90.
- B) It will decrease to 0.009.
- C) It will remain exactly unchanged at 0.9.
- D) It cannot be determined without knowing Y.
Show answer & explanation
Answer: C) It will remain exactly unchanged at 0.9.
The correlation coefficient (r) is independent of the change of scale; multiplying observations by a large constant does not alter the underlying linear value of r.
Question 28
If a data set has extreme outlier values exclusively on the higher end, how will this affect the arithmetic mean compared to the median?
- A) The mean will be pulled substantially higher than the median.
- B) The mean will be pulled substantially lower than the median.
- C) The mean and median will remain perfectly equal.
- D) The median will automatically become negative.
Show answer & explanation
Answer: A) The mean will be pulled substantially higher than the median.
In a distribution that heavily departs from symmetry and piles up at one end, extreme values will drastically drag the arithmetic mean in the direction of the skewness.
Question 29
Which specific statistical value is mathematically defined as the most frequently repeated observation in a given dataset?
- A) Arithmetic Mean
- B) Median
- C) Mode
- D) Range
Show answer & explanation
Answer: C) Mode
Unlike the arithmetic mean which averages data, the mode is specifically identified as the single most repeated value within a set of observations.
Question 30
If you subtract 3 from each observation of variable X and add 5 to each observation of variable Y, what will happen to a correlation coefficient of 0.85?
- A) It will systematically increase to 0.87.
- B) It will progressively decrease to 0.83.
- C) It will remain exactly unchanged at 0.85.
- D) It will instantly become negative.
Show answer & explanation
Answer: C) It will remain exactly unchanged at 0.85.
Adding or subtracting any constants to the independent or dependent variables represents a change of origin, which does not affect the mathematical strength or direction of the correlation.
Question 31
A fundamental mathematical property of the arithmetic mean dictates that the sum of the deviations of all individual observations from their mean will always equal:
- A) Exactly zero
- B) The standard deviation
- C) The variance
- D) The median
Show answer & explanation
Answer: A) Exactly zero
The arithmetic mean acts as the exact mathematical center of gravity for a dataset. Therefore, the sum of the positive and negative deviations from the mean (∑(x - x̄)) perfectly cancel each other out to equal zero.
Question 32
A company has two departments. Department A has 20 employees with an average salary of Rs. 50,000. Department B has 30 employees with an average salary of Rs. 60,000. What is the combined mean salary?
- A) Rs. 55,000
- B) Rs. 56,000
- C) Rs. 58,000
- D) Rs. 60,000
Show answer & explanation
Answer: B) Rs. 56,000
The combined mean is found using (n1x̄1 + n2x̄2) / (n1 + n2). Here: (20 * 50,000) + (30 * 60,000) = 1,000,000 + 1,800,000 = 2,800,000. Dividing by the total 50 employees yields Rs. 56,000.
Question 33
Which specific graphical representation tool can be utilized by statisticians to accurately estimate the median of a grouped frequency distribution?
- A) A simple bar chart
- B) A pie chart
- C) An ogive (cumulative frequency curve)
- D) A scatter diagram
Show answer & explanation
Answer: C) An ogive (cumulative frequency curve)
The median (the 50th percentile) can be visually estimated by locating N/2 on the cumulative frequency y-axis of an ogive and drawing a line to the curve, then down to the x-axis.
Question 34
Which graphical charting tool is typically utilized to visibly locate the 'mode' of a grouped continuous frequency distribution?
- A) Histogram
- B) Frequency Polygon
- C) Ogive
- D) Box and Whisker Plot
Show answer & explanation
Answer: A) Histogram
The mode is the value with the highest frequency. In a histogram, the mode can be estimated by drawing diagonal lines inside the tallest bar connecting its top corners to the adjacent bars.
Question 35
In a moderately skewed statistical distribution, which empirical relationship accurately links the mean, median, and mode?
- A) Mean - Mode = 3(Mean - Median)
- B) Mean = Median = Mode
- C) Mode = 3(Mean) - 2(Median)
- D) Median = 2(Mean - Mode)
Show answer & explanation
Answer: A) Mean - Mode = 3(Mean - Median)
For moderately skewed distributions, Karl Pearson's empirical formula states that the distance between the mean and the mode is approximately three times the distance between the mean and the median.
Question 36
If a constant value of 8 is mathematically added to every single observation in a dataset, what will be the resulting effect on the dataset's variance?
- A) It will increase by 8.
- B) It will increase by 64.
- C) It will strictly remain unchanged.
- D) It will be multiplied by 8.
Show answer & explanation
Answer: C) It will strictly remain unchanged.
Variance measures the spread or dispersion of data. Adding or subtracting a constant merely shifts the entire dataset (change of origin) without altering how spread out the individual points are from one another.
Question 37
If every observation in a sample is multiplied by 4, what will happen to the calculated standard deviation?
- A) It will remain unchanged.
- B) It will be multiplied by 4.
- C) It will be multiplied by 16.
- D) It will increase by exactly 4 units.
Show answer & explanation
Answer: B) It will be multiplied by 4.
Standard deviation is heavily affected by a change of scale. If every value is multiplied by a constant 'k', the standard deviation of the new set will be multiplied by the absolute value of 'k'.
Question 38
An investment portfolio yields an average annual return of 12% with a standard deviation of 3%. What is the portfolio's Coefficient of Variation (CV)?
- A) 25%
- B) 36%
- C) 400%
- D) 0.25%
Show answer & explanation
Answer: A) 25%
The Coefficient of Variation (CV) measures relative dispersion and is calculated as (Standard Deviation / Mean) * 100. Here, (3 / 12) * 100 = 0.25 * 100 = 25%.
Question 39
If a dataset features a long, drawn-out tail of extreme outlier values strictly on the higher (right) side, the distribution is considered:
- A) Perfectly symmetrical
- B) Negatively skewed
- C) Positively skewed
- D) Uniformly distributed
Show answer & explanation
Answer: C) Positively skewed
A distribution with a long tail stretching to the right (toward higher positive numbers) is termed positively skewed. In this scenario, the extreme high values pull the mean higher than the median.
Question 40
When attempting to calculate the true average speed of a vehicle that travels a constant distance at multiple different speeds, which statistical mean is mathematically appropriate?
- A) The Arithmetic Mean
- B) The Geometric Mean
- C) The Harmonic Mean
- D) The Weighted Mean
Show answer & explanation
Answer: C) The Harmonic Mean
The harmonic mean is specifically designed for situations dealing with rates and ratios, such as calculating the true average speed of journeys spanning equal distances.
Question 41
What does the Third Quartile (Q3) of a dataset mathematically correspond to?
- A) The 25th Percentile
- B) The 50th Percentile (Median)
- C) The 75th Percentile
- D) The absolute maximum value
Show answer & explanation
Answer: C) The 75th Percentile
Quartiles divide data into four equal parts. The third quartile (Q3) marks the point where 75% of the data values lie strictly below it, making it identical to the 75th percentile.
Question 42
Which of the following is the correct formula to calculate the Semi-Interquartile Range (often termed Quartile Deviation)?
- A) Q3 - Q1
- B) (Q3 - Q1) / 2
- C) (Q3 + Q1) / 2
- D) Q3 / Q1
Show answer & explanation
Answer: B) (Q3 - Q1) / 2
The Interquartile Range is Q3 - Q1. The prefix 'semi' mathematically indicates that this value is halved, resulting in the standard formula (Q3 - Q1) / 2.
Question 43
What is the median of the following ungrouped dataset featuring an even number of observations: 14, 18, 22, 26?
- A) 18
- B) 20
- C) 22
- D) 24
Show answer & explanation
Answer: B) 20
For an even number of observations sorted in ascending order, the median is strictly the arithmetic average of the two central values. Here, (18 + 22) / 2 = 40 / 2 = 20.
Question 44
To properly average sequential rates of compounding growth over multiple years (such as population growth or compound interest), a statistician must use the:
- A) Harmonic Mean
- B) Geometric Mean
- C) Arithmetic Mean
- D) Median
Show answer & explanation
Answer: B) Geometric Mean
The geometric mean, found by multiplying 'n' growth values together and taking the nth root, is the only statistically accurate method to average figures that are actively compounding over time.
Question 45
Which statistical measure of relative dispersion is utilized to compare the intrinsic volatility of two different datasets that are measured in completely different units (e.g., comparing weight in kg to height in cm)?
- A) Standard Error
- B) Absolute Variance
- C) Coefficient of Variation
- D) Interquartile Range
Show answer & explanation
Answer: C) Coefficient of Variation
The Coefficient of Variation (CV) divides the standard deviation by the mean and expresses the result as a pure percentage, making it a unitless measure perfect for comparing wildly different datasets.
Question 46
According to the fundamental properties of the arithmetic mean, what does the sum of the deviations of all observations from their mathematical mean precisely equal?
- A) Exactly zero
- B) The variance
- C) The standard deviation
- D) 100%
Show answer & explanation
Answer: A) Exactly zero
The arithmetic mean serves as the absolute center of gravity for the data. Consequently, the sum of all positive and negative deviations from the mean (∑(x - x̄)) mathematically perfectly cancel out to equal zero.
Question 47
A retail branch with 40 staff members pays an average salary of Rs. 45,000. Another branch with 60 staff members pays an average salary of Rs. 55,000. What is the combined mean salary of both branches?
- A) Rs. 50,000
- B) Rs. 51,000
- C) Rs. 52,000
- D) Rs. 53,000
Show answer & explanation
Answer: B) Rs. 51,000
The combined mean is (n1x̄1 + n2x̄2) / (n1 + n2). Here: (40 * 45,000) + (60 * 55,000) = 1,800,000 + 3,300,000 = 5,100,000. Dividing by the total 100 staff members yields Rs. 51,000.
Question 48
If a constant value of 15 is systematically subtracted from every observation in a dataset, what is the resulting effect on the dataset's standard deviation?
- A) It decreases by exactly 15 units.
- B) It is divided by 15.
- C) It remains completely unchanged.
- D) It decreases by the square root of 15.
Show answer & explanation
Answer: C) It remains completely unchanged.
Standard deviation measures the relative dispersion (spread) of data points around the mean. Subtracting a constant strictly shifts the entire distribution's origin without altering the physical spread of the points, leaving dispersion unchanged.
Question 49
What is the median of the following ungrouped dataset containing an even number of observations: 12, 16, 24, 30?
- A) 16
- B) 20
- C) 24
- D) 28
Show answer & explanation
Answer: B) 20
For an even number of sequentially ordered observations, the median is the arithmetic average of the two central values. Here, (16 + 24) / 2 = 40 / 2 = 20.
Question 50
In a moderately skewed frequency distribution, Karl Pearson's empirical relationship between the Mean, Median, and Mode is mathematically defined as:
- A) Mode = 3(Median) - 2(Mean)
- B) Median = 2(Mean) - Mode
- C) Mean = 3(Mode) - 2(Median)
- D) Mean = Median = Mode
Show answer & explanation
Answer: A) Mode = 3(Median) - 2(Mean)
Pearson's empirical formula states that Mean - Mode = 3(Mean - Median). By rearranging this algebraically, we get the frequently utilized formulation: Mode = 3(Median) - 2(Mean).
Question 51
An asset generates an average return of 20% with a standard deviation of 5%. What is the precisely calculated Coefficient of Variation (CV)?
- A) 4%
- B) 15%
- C) 25%
- D) 400%
Show answer & explanation
Answer: C) 25%
The Coefficient of Variation measures relative dispersion as a percentage. It is calculated as (Standard Deviation / Mean) * 100. Here, (5 / 20) * 100 = 0.25 * 100 = 25%.
Question 52
To properly calculate the average annual compounding rate of inflation over a consecutive ten-year period, a statistician must strictly utilize the:
- A) Arithmetic Mean
- B) Geometric Mean
- C) Harmonic Mean
- D) Weighted Median
Show answer & explanation
Answer: B) Geometric Mean
The geometric mean is uniquely mathematically equipped to calculate the true average of sequential compounding growth rates (like inflation or compound interest), preventing the distortion that the arithmetic mean would cause.
Question 53
When calculating the exact average speed of a train that travels multiple distinct, equal-distance segments at different speeds, which measure of central tendency is required?
- A) Arithmetic Mean
- B) Geometric Mean
- C) Harmonic Mean
- D) Standard Deviation
Show answer & explanation
Answer: C) Harmonic Mean
The harmonic mean is specifically designed for statistical situations dealing with rates, ratios, and relative velocities, such as calculating the true average speed across equal distance segments.
Question 54
If a dataset's arithmetic mean is strictly lower than its median, and the median is strictly lower than the mode (Mean < Median < Mode), the distribution is visually:
- A) Perfectly symmetrical
- B) Positively skewed
- C) Negatively skewed
- D) Uniformly flat
Show answer & explanation
Answer: C) Negatively skewed
A negatively skewed distribution has a long tail stretching to the left (towards lower values). These extreme low outliers pull the mathematical mean down so that it falls strictly below both the median and the mode.
Question 55
Which of the following graphical tools is generally utilized to visually estimate the Median of a grouped frequency distribution?
- A) A Histogram
- B) A Pie Chart
- C) An Ogive (Cumulative Frequency Curve)
- D) A Scatter Diagram
Show answer & explanation
Answer: C) An Ogive (Cumulative Frequency Curve)
The median (the exact 50th percentile) can be visually estimated by locating N/2 on the y-axis of a cumulative frequency ogive, tracing horizontally to the curve, and then tracing vertically down to the x-axis.
Question 56
If every single observation in a dataset is systematically multiplied by a constant factor of 3, what mathematical effect will this have on the arithmetic mean?
- A) It will remain unchanged.
- B) It will increase by 3 units.
- C) It will be multiplied by exactly 3.
- D) It will be divided by 3.
Show answer & explanation
Answer: C) It will be multiplied by exactly 3.
The arithmetic mean is highly sensitive to a change of scale. If every observation is multiplied by a constant, the resulting mean will be strictly multiplied by that exact same constant.
Question 57
What is the standard formula utilized to calculate the Semi-Interquartile Range (Quartile Deviation)?
- A) Q3 - Q1
- B) (Q3 - Q1) / 2
- C) (Q3 + Q1) / 2
- D) Q3 / Q1
Show answer & explanation
Answer: B) (Q3 - Q1) / 2
The regular Interquartile Range is calculated as Q3 - Q1. The prefix 'semi' mathematically dictates that this total distance is halved, giving the standard formula (Q3 - Q1) / 2.
Question 58
Which statistical measure of central tendency is formally defined as the single value that appears with the highest frequency within a dataset?
- A) Mean
- B) Median
- C) Mode
- D) Variance
Show answer & explanation
Answer: C) Mode
In descriptive statistics, the mode is defined strictly as the most typical or most frequently repeated value occurring within a specific dataset.
Question 59
If all values sampled in a dataset are perfectly identical (e.g., 10, 10, 10, 10), what is the mathematical value of the variance?
- A) Exactly zero
- B) Exactly 10
- C) Undefined
- D) Infinity
Show answer & explanation
Answer: A) Exactly zero
Variance mathematically measures the dispersion or spread of data points away from the mean. If every single observation is completely identical, there is zero spread, meaning variance strictly equals zero.
Question 60
Which specific statistical property dictates that the sum of the squared deviations of observations is mathematically minimized?
- A) When taken from the median.
- B) When taken from the mode.
- C) When taken from the arithmetic mean.
- D) When taken from the origin.
Show answer & explanation
Answer: C) When taken from the arithmetic mean.
A core mathematical property of the arithmetic mean is that the sum of the squared deviations (∑(x - x̄)²) is always an absolute minimum when calculated directly from the mean compared to any other value.