Chapter 26: Statistics and Measures of Dispersion (Set-3)
A data set has 9 values; the median position is
A 4th value
B 6th value
C 9th value
D 5th value
For odd n, the median is at position (n+1)/2 after arranging data. With n=9, (9+1)/2=5, so the 5th observation is the median.
A data set has 10 values; median is the average of
A 4th and 5th
B 6th and 7th
C 5th and 6th
D 1st and 10th
When n is even, median is the mean of the two central values after sorting. For n=10, central positions are 5 and 6, so median is average of 5th and 6th.
For 2, 2, 3, 8, 10 the mean is
A 5
B 4
C 6
D 3
Mean = (2+2+3+8+10)/5 = 25/5 = 5. Mean uses every observation, so it shifts toward higher values compared to the median in skewed sets.
For 1, 2, 2, 9, 10 the median is
A 4
B 2
C 1
D 9
Sorted data are 1,2,2,9,10. The middle (3rd) value is 2, so median is 2. Median is resistant to extremes like 9 and 10.
For 4, 4, 6, 7, 9 the mode is
A 6
B 7
C 4
D 9
Mode is the most frequent value. Here 4 occurs twice while others occur once, so 4 is the mode. Mode is useful even when data are categorical.
If mean = 12 and median = 10, then (approx.) mode is
A 6
B 8
C 14
D 16
Mode ≈ 3Median − 2Mean = 30 − 24 = 6. This approximation works for moderately skewed distributions and helps estimate one central measure from the other two.
In grouped data, the class with highest frequency is
A Median class
B Mean class
C Modal class
D Quartile class
The modal class is the class interval having the largest frequency. Grouped mode is estimated using the modal class and its adjacent class frequencies.
In grouped median formula, N represents
A Total frequency
B Class width
C Cumulative frequency
D Class midpoint
In grouped data, N is the total number of observations, equal to sum of all frequencies. Median class is found using N/2 in the cumulative frequency column.
For grouped median, CF before median class is
A Frequency of median class
B Cumulative frequency previous
C Total frequency N
D Upper boundary
The grouped median formula uses cumulative frequency of the class just before the median class. It helps measure how many observations lie below the median class boundary.
Range coefficient (relative) is
A (L+S)/(L−S)
B (L−S)/2
C (L−S)/(L+S)
D (L+S)/2
Coefficient of range compares spread relative to extremes, where L is largest and S is smallest. It is unit-free and useful for comparing variability across datasets.
Interquartile range mainly covers
A Middle 50% data
B Middle 25% data
C Entire data range
D Only extremes
IQR = Q3 − Q1 represents the spread of the central half of the data. It ignores the lowest 25% and highest 25%, reducing impact of outliers.
If Q1 = 12 and Q3 = 28, quartile deviation is
A 16
B 20
C 40
D 8
Quartile deviation = (Q3 − Q1)/2 = (28 − 12)/2 = 16/2 = 8. It measures typical spread of middle half of the data.
Mean deviation about mean uses
A (x−x̄) values
B (x−x̄)² values
C |x−x̄| values
D x² values
Mean deviation uses absolute deviations so positive and negative deviations do not cancel. It provides an average distance from the mean but is less algebra-friendly than variance.
Mean deviation coefficient about median is
A Median/MD(Median)
B MD(Median)/Median
C MD(Mean)/Mean
D Mean/MD(Mean)
Coefficient of mean deviation is a relative measure: mean deviation divided by the central value used. About median, it is MD about median divided by median.
For ungrouped data, variance equals
A Mean of squared deviations
B Mean of absolute deviations
C Median of deviations
D Range of deviations
Variance is the average of (x−mean)². Squaring avoids sign cancellation and gives higher weight to larger deviations, which is important in many theoretical results.
If variance is 2.25, SD equals
A 2.25
B 4.5
C 1.5
D 0.5
Standard deviation is the square root of variance. √2.25 = 1.5. SD is expressed in the original unit, making spread interpretation easier.
If SD = 6, variance is
A 36
B 12
C 6
D 3
Variance is the square of SD. If SD = 6, variance = 6² = 36. Variance is in squared units, so it’s less intuitive but useful in calculations.
A dataset has mean 50 and SD 5; z-score of 60 is
A 1
B 3
C 2
D 0
z = (x − mean)/SD = (60 − 50)/5 = 10/5 = 2. So 60 is two standard deviations above the mean.
If a constant c is added to all values, mean
A Decreases by c
B Increases by c
C Becomes unchanged
D Becomes zero
Adding a constant shifts every observation upward by c, so mean also increases by c. Dispersion measures based on deviations (variance, SD) stay unchanged.
If a constant c is added to all values, median
A Decreases by c
B Becomes unchanged
C Becomes half
D Increases by c
Median is a positional measure. Adding c to every value shifts the whole ordered list by c, so the middle value also increases by c.
If all values are multiplied by 2, mean becomes
A Halved
B Unchanged
C Doubled
D Zero
Scaling all observations by 2 scales the mean by 2 because mean is linear. Every value doubles, so their average also doubles.
If all values are multiplied by 2, SD becomes
A Doubled
B Halved
C Unchanged
D Squared
Standard deviation scales by the absolute value of the multiplier. Multiply data by 2 ⇒ deviations double ⇒ SD doubles. Variance would become four times.
If all values are multiplied by 2, variance becomes
A Two times
B Four times
C Unchanged
D Half
Variance scales with the square of the multiplier. Multiply by 2 ⇒ deviations multiply by 2 ⇒ squared deviations multiply by 4 ⇒ variance becomes four times.
A set has larger SD; it indicates
A Greater spread
B Smaller spread
C Same spread
D No data
Standard deviation measures typical distance from the mean. Larger SD means values are more scattered around the mean, showing higher variability.
Comparing two series with different units, best measure is
A Standard deviation
B Variance
C Coefficient of variation
D Range
CV is unit-free because it’s SD divided by mean. This makes it suitable for comparing relative variability across series with different units or different mean levels.
If Series A CV < Series B CV, Series A is
A Less consistent
B More skewed
C Always normal
D More consistent
Lower CV means smaller spread relative to mean. Hence values are more tightly clustered, indicating greater consistency and reliability compared to the series with higher CV.
For data 1, 3, 5, 7, mean deviation about mean is
A 2
B 1
C 4
D 0
Mean is 4. Absolute deviations: |1−4|=3, |3−4|=1, |5−4|=1, |7−4|=3. Sum=8, divide by 4 gives 2.
For data 2, 4, 6, 8, variance (population) is
A 4
B 2
C 5
D 10
Mean is 5. Deviations: −3, −1, 1, 3; squares: 9,1,1,9 sum=20. Population variance = 20/4 = 5.
For data 2, 4, 6, 8, SD (population) is
A 5
B √5
C 2
D √20
From previous, population variance = 5. SD is √variance = √5. SD is in original unit, showing typical distance from mean.
If mean = 30 and CV = 10%, SD is
A 10
B 0.3
C 3
D 300
CV = (SD/Mean)×100. So SD = (CV/100)×Mean = (10/100)×30 = 3. This gives relative spread around mean.
Standard error becomes smaller when
A Sample size increases
B Mean increases only
C Range increases
D Variance becomes zero
Standard error of mean is roughly SD/√n. As sample size n increases, √n increases, so standard error decreases, meaning sample mean becomes more stable.
A negatively skewed distribution generally has
A Mean > median
B Mean < median
C Mean = median
D Mode < mean
In negative skew, tail extends to the left, pulling mean leftward. Typically mean < median < mode. This is a common pattern for moderately skewed distributions.
A positively skewed distribution generally has
A Mean < median
B Mean = mode
C Mean > median
D Median > mode
In positive skew, right tail is longer, pulling mean rightward. Commonly mode < median < mean. This helps judge skewness direction from central tendency measures.
When plotting a histogram, rectangles touch because
A Data are categories
B Data are random
C Data are bivariate
D Data are continuous
Histograms represent continuous class intervals without gaps. Since classes cover continuous ranges, bars touch. Bar charts for categorical data have gaps between bars.
Frequency density is needed when
A Unequal class widths
B Equal class widths
C No class intervals
D Only raw values
When class widths differ, bar heights should represent frequency per unit width. Frequency density = frequency/class width, ensuring histogram area reflects frequency correctly.
In a histogram with unequal widths, area represents
A Mean
B SD
C Frequency
D Median
For unequal class intervals, the area of each rectangle (height × width) represents frequency. Using frequency density as height preserves correct frequency comparison.
A measure useful for “middle spread” is
A Quartile deviation
B Range
C Mean
D Mode
Quartile deviation depends on Q1 and Q3, focusing on central 50% of data. It ignores extremes, making it useful when outliers exist and middle spread matters.
Variance cannot be
A Zero
B Positive
C Fractional
D Negative
Variance is the average of squared deviations, and squares are never negative. It can be zero (all values equal) or positive, and can be fractional.
The SD of a dataset is zero when
A Values are negative
B All values equal
C Mean is zero
D Range is large
SD measures spread around mean. If all values are identical, every deviation is zero. Hence variance and SD become zero, showing no variability.
For grouped mean using assumed mean method, we use
A Deviations from assumed mean
B Only class boundaries
C Cumulative frequency
D Only extremes
Assumed mean method simplifies calculation by taking a convenient A and computing deviations (d = x − A) using class marks. It reduces arithmetic load for grouped data.
Step-deviation method is most helpful when
A Data are categorical
B Only two values
C Class width common
D No frequencies
Step-deviation uses u = (x−A)/h when class width h is common. This reduces large numbers and makes computing mean and SD in grouped data faster.
In step-deviation, h represents
A Common class width
B Highest frequency
C Cumulative frequency
D Median position
In step-deviation method, h is the common class interval size. Dividing by h scales deviations into smaller integers, simplifying computations for grouped statistics.
If covariance is positive, variables tend to
A Move opposite
B Increase together
C Have no link
D Be identical
Positive covariance indicates that when one variable is above its mean, the other tends to be above its mean too. It shows direction of joint movement.
Correlation is preferred over covariance because
A Always positive
B Uses only medians
C Unit-free measure
D Ignores sample size
Correlation standardizes covariance by dividing by SDs, removing units. This allows meaningful comparison of strength of association across different datasets and measurement scales.
A good use of ogive is finding
A Quartiles graphically
B Mode directly
C Variance quickly
D Range only
Ogives help locate median, quartiles, and percentiles using cumulative frequency curves. You can draw horizontal lines at required cumulative values and read corresponding x-values.
If the distribution is symmetric, usually
A Mean less median
B Median less mean
C Mode not exists
D Mean equals median
In symmetric distributions, the center is balanced on both sides. Mean and median coincide, and for a perfectly symmetric unimodal distribution, mode also matches them.
If mean is greater than median, skewness is likely
A Negative skew
B Positive skew
C Zero skew
D Bimodal
Mean is pulled toward the tail. If mean > median, the tail is typically on the right side, indicating positive (right) skewness for moderately skewed data.
A dataset where mode is not defined usually is
A All values different
B All values same
C Two values same
D Only grouped data
If every value occurs only once, there is no most frequent value, so mode is not defined. If all values are identical, that value is the mode.
If two series have same SD but different means, compare consistency using
A Range
B Variance
C Coefficient of variation
D Mean deviation
When means differ, SD alone is not enough for relative consistency. CV adjusts spread relative to mean, allowing fair comparison of variability in such situations.
The main purpose of dispersion measures is to
A Find central value
B Count observations
C Label categories
D Quantify spread
Dispersion measures describe how far observations scatter around a central value. They complement measures of central tendency and help compare stability, consistency, and reliability of datasets.