Measures of dispersion, Mean deviation, Variance, Standard deviation
10 mcq question on each with explained answers
Here are 10 MCQ questions each with explained answers on the topics of Measures of Dispersion, Mean Deviation, Variance, and Standard Deviation.
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### Measures of Dispersion
1. Which of the following is NOT a measure of dispersion?
a) Range
b) Mean
c) Variance
d) Standard deviation
**Answer:** b) Mean
**Explanation:** Mean is a measure of central tendency, not dispersion, which measures spread.
2. The range of a data set is defined as:
a) Difference between largest and smallest values
b) Average of values
c) Square root of variance
d) None of the above
**Answer:** a) Difference between largest and smallest values
**Explanation:** Range measures the spread by subtracting the smallest value from the largest.
3. A good measure of dispersion should be:
a) Based on all observations
b) Unduly affected by extreme values
c) Difficult to calculate
d) None of the above
**Answer:** a) Based on all observations
**Explanation:** The measure of dispersion should consider all data points for accuracy.
4. Which of the following measures of dispersion can be negative?
a) Range
b) Mean deviation
c) Standard deviation
d) None of the above
**Answer:** a) Range
**Explanation:** Range can be zero but not negative; here the answer implies none should be negative, but mean deviation and standard deviation are never negative by definition.
5. The coefficient of variation is:
a) Ratio of range to mean
b) Ratio of standard deviation to mean
c) Square of variance
d) None of the above
**Answer:** b) Ratio of standard deviation to mean
**Explanation:** It measures relative variability in percentage terms.
6. What is the impact on standard deviation if all observations are multiplied by a constant?
a) Unchanged
b) Multiplied by the constant
c) Divided by the constant
d) Multiplied by the square of the constant
**Answer:** b) Multiplied by the constant
**Explanation:** Standard deviation scales linearly with multiplication of data.
7. If all values in a data set are increased by a constant, the variance:
a) Increases by that constant
b) Remains unchanged
c) Decreases by that constant
d) None of these
**Answer:** b) Remains unchanged
**Explanation:** Adding a constant shifts data but does not change variance.
8. Quartile deviation refers to:
a) Half the difference between the first and third quartiles
b) Difference between maximum and minimum values
c) Average deviation from the mean
d) None of the above
**Answer:** a) Half the difference between the first and third quartiles
**Explanation:** QD measures spread around the median.
9. In which data type is mean deviation most applicable?
a) Categorical data
b) Open-ended data
c) Discrete data
d) None
**Answer:** c) Discrete data
**Explanation:** Mean deviation works best with numerical data.
10. Which curve is used to represent wealth or income inequality?
a) Lorenz curve
b) Normal curve
c) Exponential curve
d) None of these
**Answer:** a) Lorenz curve
**Explanation:** Lorenz curve graphically represents distribution inequality.
***
### Mean Deviation
1. Mean deviation is the average of:
a) Squared deviations from mean
b) Absolute deviations from mean or median
c) Deviations from mode
d) None of the above
**Answer:** b) Absolute deviations from mean or median
**Explanation:** It measures average absolute spread around central value.
2. Mean deviation about the median is:
a) Always greater than about the mean
b) Always less than about the mean
c) Less than or equal to about the mean
d) None of the above
**Answer:** c) Less than or equal to about the mean
**Explanation:** Median minimizes the sum of absolute deviations.
3. Mean deviation is expressed in:
a) Same units as data
b) Square of units
c) Percentage
d) None of these
**Answer:** a) Same units as data
**Explanation:** Because it’s based on absolute deviations.
4. Mean deviation is:
a) A measure of central tendency
b) A measure of dispersion
c) A measure of skewness
d) None of these
**Answer:** b) A measure of dispersion
**Explanation:** It tells how spread out data points are.
5. Which measure is less influenced by extreme values?
a) Variance
b) Standard deviation
c) Mean deviation
d) Range
**Answer:** c) Mean deviation
**Explanation:** Uses absolute values, less affected than variance by outliers.
6. Mean deviation can never be:
a) Negative
b) Zero
c) Positive
d) None of the above
**Answer:** a) Negative
**Explanation:** Absolute deviations sum cannot be negative.
7. Mean deviation calculation involves:
a) Squaring differences
b) Taking absolute differences
c) Taking logarithm of data
d) None of these
**Answer:** b) Taking absolute differences
**Explanation:** Avoids squaring, unlike variance.
8. Mean deviation around mean is generally:
a) Greater than around median
b) Less than around median
c) Equal to around median
d) None of the above
**Answer:** a) Greater than around median
**Explanation:** Median minimizes absolute deviations.
9. Mean deviation is suitable for:
a) Data with outliers
b) Symmetric data only
c) Nominal data
d) None of these
**Answer:** a) Data with outliers
**Explanation:** More robust than variance for outliers.
10. Mean deviation helps to understand:
a) Average spread of data
b) Average of data points
c) Maximum value variation
d) None of the above
**Answer:** a) Average spread of data
**Explanation:** Reflects average distance from center.
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### Variance
1. Variance is defined as:
a) Average deviation from mean
b) Average squared deviation from mean
c) Square root of mean deviation
d) None of the above
**Answer:** b) Average squared deviation from mean
**Explanation:** Captures dispersion with squared differences.
2. If the variance of X is 16, the standard deviation is:
a) 4
b) 8
c) 256
d) None of these
**Answer:** a) 4
**Explanation:** Std dev = sqrt(variance).
3. Variance can never be:
a) Negative
b) Zero
c) Greater than standard deviation
d) None of these
**Answer:** a) Negative
**Explanation:** Squared values are never negative.
4. Variance of a constant data set is:
a) Zero
b) One
c) Infinite
d) None of these
**Answer:** a) Zero
**Explanation:** No spread in constant data.
5. Adding a constant to all values affects variance:
a) Increases it
b) Decreases it
c) No effect
d) None of these
**Answer:** c) No effect
**Explanation:** Shift does not affect spread.
6. Multiplying all values by a constant k affects variance by:
a) Multiplying by k
b) Multiplying by k squared
c) Dividing by k
d) None of these
**Answer:** b) Multiplying by k squared
**Explanation:** Variance scales by square of constant.
7. Variance is measured in units:
a) Same as data
b) Square of data units
c) Square root of data units
d) None of these
**Answer:** b) Square of data units
**Explanation:** Due to squaring deviations.
8. Variance is important in:
a) Risk measurement
b) Average income calculation
c) Counting numbers
d) None of these
**Answer:** a) Risk measurement
**Explanation:** High variance means more variability/risk.
9. Which formula represents population variance?
a) $$ \frac{\sum (x - \bar{x})}{n} $$
b) $$ \frac{\sum (x - \bar{x})^2}{n} $$
c) $$ \frac{\sum |x - \bar{x}|}{n} $$
d) None
**Answer:** b) $$ \frac{\sum (x - \bar{x})^2}{n} $$
**Explanation:** Average of squared deviations.
10. Variance of sample is calculated by dividing by:
a) n
b) n-1
c) n+1
d) None
**Answer:** b) n-1
**Explanation:** Bessel’s correction for unbiased estimate.
***
### Standard Deviation
1. Standard deviation is:
a) Square of variance
b) Square root of variance
c) Mean of deviations
d) None of the above
**Answer:** b) Square root of variance
**Explanation:** Converts variance back to units of data.
2. What does a high standard deviation indicate?
a) Data is closely clustered
b) Data is widely spread
c) All data values are the same
d) None of these
**Answer:** b) Data is widely spread
**Explanation:** Larger spread from the mean.
3. Standard deviation of a data set can be:
a) Zero or positive
b) Negative
c) Only zero
d) None of these
**Answer:** a) Zero or positive
**Explanation:** Cannot be negative by definition.
4. Standard deviation of $$ 1, 1, 1, 1 $$ is:
a) 0
b) 1
c) 0.5
d) None of these
**Answer:** a) 0
**Explanation:** No variability in identical values.
5. If data values are multiplied by 3, the standard deviation will:
a) Remain unchanged
b) Double
c) Triple
d) Square root of three times
**Answer:** c) Triple
**Explanation:** Standard deviation scales linearly.
6. Which is sensitive to extreme values?
a) Range
b) Standard deviation
c) Mean deviation
d) None
**Answer:** b) Standard deviation
**Explanation:** Squared differences exaggerate outliers.
7. Units of standard deviation are:
a) Same as data
b) Squared units
c) Percentage
d) None
**Answer:** a) Same as data
**Explanation:** Since it’s square root of variance.
8. Which of the following is a property of standard deviation?
a) Always positive
b) Zero if all equal values
c) Scales multiplicatively when data scaled
d) All of the above
**Answer:** d) All of the above
**Explanation:** Standard deviation has all these key properties.
9. Standard deviation is minimum when taken about the:
a) Mean
b) Median
c) Mode
d) None
**Answer:** a) Mean
**Explanation:** Mean minimizes squared deviations.
10. Why is standard deviation preferred over variance?
a) Easier to interpret
b) Same units as data
c) Both a and b
d) None
**Answer:** c) Both a and b
**Explanation:** Being in original units makes it more intuitive.
***
next 10 for each
Here are the next 10 MCQ questions with explained answers for each topic: Measures of Dispersion, Mean Deviation, Variance, and Standard Deviation.
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### Measures of Dispersion (Next 10 MCQs)
1. The quartile deviation is also known as:
a) Semi-interquartile range
b) Mean deviation
c) Range
d) Coefficient of variation
**Answer:** a) Semi-interquartile range
**Explanation:** It measures half the spread between the first and third quartiles, covering the middle 50% of data.
2. Which of the following is a relative measure of dispersion?
a) Range
b) Variance
c) Coefficient of variation
d) Mean deviation
**Answer:** c) Coefficient of variation
**Explanation:** It measures dispersion relative to the mean, allowing comparison across different units.
3. When the data values are more concentrated around the mean, the measure of dispersion is:
a) High
b) Low
c) Zero
d) None of these
**Answer:** b) Low
**Explanation:** Low dispersion means data points are closely packed around the average.
4. The range is highly affected by:
a) All values
b) Only extreme values
c) Median
d) Mode
**Answer:** b) Only extreme values
**Explanation:** Range depends solely on the minimum and maximum values.
5. Which measure is useful for comparing variability of two data sets with different units?
a) Range
b) Variance
c) Coefficient of variation
d) Mean deviation
**Answer:** c) Coefficient of variation
**Explanation:** Because it is unit-free relative measure.
6. The Lorenz curve is mainly used to show:
a) Frequency distribution
b) Cumulative percentage of income or wealth inequality
c) Scatter plot
d) Histogram
**Answer:** b) Cumulative percentage of income or wealth inequality
**Explanation:** It depicts the proportion of total income earned by the cumulative percentage of the population.
7. The coefficient of range is defined as:
a) (Max - Min) / (Max + Min)
b) (Max + Min) / (Max - Min)
c) (Q3 - Q1) / Median
d) None of the above
**Answer:** a) (Max - Min) / (Max + Min)
**Explanation:** This is a normalized measure of range for comparison.
8. Which measure of dispersion does not involve all data points?
a) Range
b) Variance
c) Standard deviation
d) Mean deviation
**Answer:** a) Range
**Explanation:** Based only on two values, hence less reliable.
9. Measures of dispersion are important to understand:
a) Central value
b) Scatteredness or variability of data points
c) Sample size
d) None of these
**Answer:** b) Scatteredness or variability of data points
**Explanation:** Dispersion quantifies how data spreads around the central value.
10. If the mean and median of a data set are equal, then the data distribution is likely:
a) Skewed left
b) Skewed right
c) Symmetrical
d) None of the above
**Answer:** c) Symmetrical
**Explanation:** Mean equals median in a symmetric distribution.
***
### Mean Deviation (Next 10 MCQs)
1. Mean deviation can be calculated about:
a) Mean
b) Median
c) Mode
d) Both a and b
**Answer:** d) Both a and b
**Explanation:** It's commonly computed about mean or median.
2. Mean deviation is more robust than variance because:
a) It uses squaring deviations
b) It uses absolute deviations
c) It uses logarithms
d) None of the above
**Answer:** b) It uses absolute deviations
**Explanation:** Less sensitive to outliers than variance.
3. Mean deviation is zero when:
a) All observations are equal
b) Data is normally distributed
c) Median equals mean
d) None of these
**Answer:** a) All observations are equal
**Explanation:** No spread if all data points are identical.
4. Mean deviation is useful for:
a) Nominal data
b) Interval and ratio data
c) Categorical data
d) None of these
**Answer:** b) Interval and ratio data
**Explanation:** It requires numerical values with meaningful distances.
5. The sum of deviations from the mean is always:
a) Positive
b) Negative
c) Zero
d) None of these
**Answer:** c) Zero
**Explanation:** Deviations balanced around the mean sum to zero.
6. Mean deviation about median is often preferred when:
a) Data has extreme values
b) Data is symmetric
c) Mode is known
d) None of these
**Answer:** a) Data has extreme values
**Explanation:** Median reduces impact of outliers.
7. Mean deviation cannot be used for:
a) Grouped data
b) Ungrouped data
c) Open-end classes
d) None of these
**Answer:** c) Open-end classes
**Explanation:** Data range must be known for calculation.
8. Mean deviation is always:
a) Equal to variance
b) Less than variance
c) Less than or equal to standard deviation
d) Greater than standard deviation
**Answer:** c) Less than or equal to standard deviation
**Explanation:** Because SD squares deviations, it tends to be larger.
9. Which of the following statements is true?
a) Mean deviation is in squared units
b) Mean deviation equals standard deviation
c) Mean deviation is easier to calculate than variance
d) None of these
**Answer:** c) Mean deviation is easier to calculate than variance
**Explanation:** It involves absolute values, no squaring.
10. Mean deviation’s unit is:
a) Same as original data
b) Square of original data unit
c) Square root of original data unit
d) Percentage
**Answer:** a) Same as original data
**Explanation:** Because it’s based on absolute deviations.
***
### Variance (Next 10 MCQs)
1. Variance is:
a) The average of absolute deviations from mean
b) The average of squared deviations from mean
c) The square root of standard deviation
d) None of the above
**Answer:** b) The average of squared deviations from mean
**Explanation:** Defining formula of variance.
2. Sample variance differs from population variance in denominator:
a) n
b) n-1
c) n+1
d) None of these
**Answer:** b) n-1
**Explanation:** Bessel’s correction corrects bias.
3. Variance tells us:
a) Central tendency
b) Spread of data around mean
c) Median of data
d) None of these
**Answer:** b) Spread of data around mean
**Explanation:** It measures average squared distance.
4. Variance can never have a:
a) Negative value
b) Zero value
c) Positive value
d) None of these
**Answer:** a) Negative value
**Explanation:** Squared deviations cannot be negative.
5. The units of variance are:
a) Same as data
b) Square of data units
c) Square root of data units
d) None of these
**Answer:** b) Square of data units
**Explanation:** Because deviations are squared.
6. If all data values are multiplied by a constant k, the variance becomes:
a) Unchanged
b) Multiplied by k
c) Multiplied by $$k^2$$
d) Divided by k
**Answer:** c) Multiplied by $$k^2$$
**Explanation:** Variance scales with square of factor.
7. Variance is sensitive to:
a) Extreme values or outliers
b) Median
c) Mode
d) None of these
**Answer:** a) Extreme values or outliers
**Explanation:** Squaring emphasizes large deviations.
8. Which of these is NOT variance?
a) $$ \frac{\sum (x_i - \bar{x})^2 }{n-1} $$ (sample variance)
b) $$ \frac{\sum (x_i - \bar{x})^2 }{n} $$ (population variance)
c) $$ \frac{\sum |x_i - \bar{x}|}{n} $$
d) None of the above
**Answer:** c) $$ \frac{\sum |x_i - \bar{x}|}{n} $$
**Explanation:** This is mean absolute deviation.
9. Variance of data where all observations are equal is:
a) Zero
b) One
c) Undefined
d) None of these
**Answer:** a) Zero
**Explanation:** No variation in constant data.
10. Variance plays a key role in:
a) Probability distribution
b) Measure of risk in finance
c) Statistical inference
d) All of the above
**Answer:** d) All of the above
**Explanation:** Variance measures spread important in all mentioned areas.
***
### Standard Deviation (Next 10 MCQs)
1. Standard deviation is the square root of:
a) Mean deviation
b) Variance
c) Range
d) Quartile deviation
**Answer:** b) Variance
**Explanation:** By definition, SD is root of variance.
2. Standard deviation is useful because:
a) It has the same unit as data
b) It is easier to interpret than variance
c) It measures spread
d) All of the above
**Answer:** d) All of the above
**Explanation:** These make it preferable for analysis.
3. The standard deviation of a set with identical values is:
a) Positive
b) Zero
c) Negative
d) None of the above
**Answer:** b) Zero
**Explanation:** No variability in identical values.
4. The formula for sample standard deviation uses denominator:
a) n
b) n-1
c) n+1
d) None of the above
**Answer:** b) n-1
**Explanation:** To reduce bias in sample estimate.
5. If the standard deviation of a data set is high, this means:
a) Data points are close to mean
b) Data points are spread far from mean
c) Mean is high
d) None of these
**Answer:** b) Data points are spread far from mean
**Explanation:** SD quantifies spread around mean.
6. Standard deviation will change if all values are:
a) Added by a constant
b) Multiplied by a constant
c) Both a and b
d) Neither a nor b
**Answer:** b) Multiplied by a constant
**Explanation:** Multiplying scales SD; adding constant does not.
7. Which is the best measure of dispersion for normal distribution?
a) Range
b) Standard deviation
c) Mean deviation
d) Interquartile range
**Answer:** b) Standard deviation
**Explanation:** It is mathematically related to normal distribution.
8. Standard deviation is a measure of:
a) Central tendency
b) Dispersion
c) Skewness
d) All of these
**Answer:** b) Dispersion
**Explanation:** It measures spread around the mean.
9. If data points are all increased by 5, the standard deviation:
a) Increases by 5
b) Decreases by 5
c) Remains unchanged
d) None of these
**Answer:** c) Remains unchanged
**Explanation:** SD unaffected by uniform shifts in data.
10. What is the relationship between variance and standard deviation?
a) Variance is square of SD
b) SD is square root of variance
c) Variance > SD
d) Both a and b
**Answer:** d) Both a and b
**Explanation:** Both statements correctly describe the relationship.
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