Population variance is 13.2, so SD = √13.2. Standard deviation is always the square root of variance and returns spread in original unit scale.
For data 10, 12, 14, 16 the population variance is
A 4
B 6
C 5
D 8
Mean = 13. Deviations: −3, −1, 1, 3; squares: 9,1,1,9 sum=20. Population variance = 20/4 = 5.
For data 10, 12, 14, 16 the population SD is
A 5
B 2
C √20
D √5
With population variance 5, SD = √5. SD is the typical spread around mean in original units, while variance is in squared units.
For data 2, 2, 2, 6, 6 the mean is
A 3.2
B 4.0
C 3.6
D 2.8
Mean = (2+2+2+6+6)/5 = 18/5 = 3.6. Mean includes all observations, so repeated values affect it strongly.
For data 2, 2, 2, 6, 6 the mode is
A 6
B 2
C 4
D 3
Mode is most frequent. Value 2 occurs three times while 6 occurs twice, so mode is 2. Mode highlights the most common observation.
For data 2, 2, 2, 6, 6 the median is
A 3
B 6
C 4
D 2
Ordered data: 2,2,2,6,6. Middle (3rd) value is 2, so median is 2. Median is position-based and less influenced by large values.
A distribution with mean > median is usually
A Left skewed
B Symmetric
C Right skewed
D Uniform
Mean is pulled toward the longer tail. If mean exceeds median, the tail is typically to the right, indicating positive (right) skewness for many practical cases.
A distribution with mean < median is usually
A Left skewed
B Right skewed
C Symmetric
D Bimodal
In negative (left) skew, the left tail is longer and pulls the mean leftward. Thus mean is typically less than median, with mode often highest among them.
If two series have same mean, comparing spread is best by
A Mean only
B Median only
C Mode only
D SD only
When means are equal, standard deviation directly compares dispersion around the same center. Lower SD means data are closer to the mean, showing more consistency.
If two series have different means, consistency is best compared using
A SD
B Variance
C CV
D Range
Coefficient of variation compares SD relative to mean, making it fair for series with different mean levels or different units. Lower CV indicates more consistency.
For grouped mean, Σf represents
A Total frequency
B Total classes
C Class width
D Midpoint sum
In grouped calculations, Σf is the sum of frequencies and equals total number of observations. It is used in mean formula Σ(fx)/Σf and in many other measures.
For grouped mean, Σfx means
A Sum of f only
B Sum of x only
C Sum of f×x
D Sum of class widths
Σfx is the sum of frequency multiplied by corresponding class mark x. Dividing Σfx by Σf gives the grouped mean, assuming class mark represents each class.
In step-deviation mean, u is defined as
A (x+A)/h
B (x−A)/h
C (A−x)h
D (x−A)h
Step-deviation uses u = (x−A)/h, where A is assumed mean and h is common class width. This reduces calculations by using smaller numbers.
In step-deviation, grouped mean equals
A A − h(Σfu/Σf)
B h + A(Σfu/Σf)
C A(Σf/Σfu)
D A + h(Σfu/Σf)
Using u values, mean is adjusted from assumed mean A by adding h times average u: x̄ = A + h(Σfu/Σf). It is efficient when data are large.
For grouped variance using assumed mean, we need
A Σf only
B ΣCF only
C Σfd² and Σfd
D Σclass widths
In assumed mean method, d = x−A. Grouped variance can be computed using Σfd² and Σfd with correction term, then divided by Σf (population case).
If d = x−A, then (x̄ − A) equals
A Σfd² / Σf
B Σfd / Σf
C Σf / Σfd
D Σx / Σf
From grouped mean formula, x̄ = A + (Σfd/Σf). So x̄ − A = Σfd/Σf. This helps simplify variance computations with an assumed mean.
Mean deviation is preferred over SD when focus is on
A Squared distances
B Only extremes
C Only median
D Absolute distances
Mean deviation uses absolute deviations, giving average “actual distance” from center. SD uses squares, emphasizing large deviations. Choice depends on interpretation and mathematical needs.
SD is preferred over mean deviation because
A Algebra-friendly
B Ignores outliers
C Uses no mean
D Always smaller
SD is based on squared deviations, which work smoothly in algebra, calculus, and probability theory. Mean deviation uses absolute values, making many theoretical manipulations harder.
For data 4, 6, 8, 10, 12 the mean is
A 7
B 9
C 8
D 10
Mean = (4+6+8+10+12)/5 = 40/5 = 8. The mean is the balance point of data and uses all values in the set.
For data 4, 6, 8, 10, 12 the mean deviation about mean is
A 2.0
B 3.0
C 4.0
D 2.4
Mean is 8. Absolute deviations: 4,2,0,2,4 sum=12. Mean deviation = 12/5 = 2.4. It measures average absolute distance from mean.
For data 4, 6, 8, 10, 12 the population variance is
A 6
B 8
C 4
D 10
Mean is 8. Deviations: −4,−2,0,2,4; squares: 16,4,0,4,16 sum=40. Population variance = 40/5 = 8.
For data 4, 6, 8, 10, 12 the population SD is
A 8
B √40
C √8
D 4
SD is square root of variance. With variance 8, SD = √8. This is about 2.828, showing typical spread around mean in original units.
The “units” of coefficient of variation are
A No units
B Same as data
C Squared units
D Percent only
CV is SD divided by mean, so units cancel. Often it is expressed as a percentage, but fundamentally it is dimensionless, useful for comparing different datasets.
If mean is negative, CV interpretation becomes
A Always perfect
B Same as SD
C Always zero
D Problematic
CV uses mean in the denominator. If mean is negative or near zero, CV can be misleading or very large. CV is best used when mean is positive and meaningful.
When class intervals are open-ended, best central measure is
A Mean
B Variance
C Median
D SD
With open-ended classes like “80 and above,” mean is difficult because class marks are unclear. Median can still be found using cumulative frequencies and median class.
In a symmetric distribution, skewness is
A Positive
B Zero
C Negative
D Undefined
Symmetry means both sides balance equally around the center. Hence there is no tilt, and skewness is zero. In such cases mean and median are typically equal.
If covariance is zero, it implies
A Perfect association
B Same mean values
C Same SD values
D No linear association
Covariance zero suggests no linear relationship in centered units, but it does not guarantee independence. Nonlinear relationships may still exist even when covariance is zero.
Correlation is undefined when
A Mean is zero
B Median is zero
C SD is zero
D Mode is zero
Correlation uses division by product of SDs. If either variable has zero SD (constant values), correlation cannot be computed because division by zero occurs.
A standard normal variable has mean and SD
A 0 and 1
B 1 and 0
C 0 and 0
D 1 and 1
Standard normal distribution is normal with mean 0 and standard deviation 1. Any normal variable can be converted to standard normal using z = (x−μ)/σ.
Which is most suitable to show cumulative distribution
A Histogram
B Pie chart
C Bar chart
D Ogive
Ogive is a cumulative frequency curve. It directly shows how many observations fall below (or above) a value and helps estimate median, quartiles, and percentiles.
If class widths are equal, histogram height represents
A Frequency density
B Cumulative frequency
C Frequency
D Mean
With equal class widths, frequency density is proportional to frequency, so using frequency as height works correctly. Then rectangle area is also proportional to frequency.
If two series have same CV, they have
A Same relative variability
B Same mean always
C Same SD always
D Same range always
Same CV means SD/mean ratio is equal. Their spreads relative to their means are the same, even if their actual means and SDs differ.
If all values are multiplied by −2, SD becomes
A Four times
B Negative two
C Two times
D Zero
SD scales by |k|. Multiplying by −2 flips sign but absolute deviations scale by 2, so SD becomes 2 times. Variance becomes 4 times due to squaring.
If all values are multiplied by −2, variance becomes
A Two times
B Negative four
C Unchanged
D Four times
Variance scales by k². With k = −2, k² = 4, so variance becomes four times. Variance never becomes negative because it is based on squares.
The median of grouped data uses
A Class marks only
B Class boundaries
C Mode formula
D Pie sectors
Grouped median formula uses lower boundary of median class, class width, cumulative frequency before median class, and frequency of median class to interpolate the median.
The grouped mode formula uses
A CF and N/2
B Σfx only
C f1, f0, f2
D Q1 and Q3
Grouped mode uses modal class frequency f1 and adjacent frequencies f0 (previous) and f2 (next). This estimates the peak location within the modal class interval.
A positively skewed distribution has tail on
A Left side
B Both sides
C No tail
D Right side
Positive skew means distribution stretches further on the right. A few large values create a longer right tail, pulling mean to the right more than the median.
In a box plot, outliers are often beyond
A 1.5×IQR fences
B 0.5×IQR fences
C 2×Range fences
D Mean±SD only
A common rule marks outliers below Q1−1.5IQR or above Q3+1.5IQR. This uses IQR as a robust spread measure less influenced by extremes.
Chebyshev’s inequality applies to
A Only normal data
B Any distribution
C Only symmetric data
D Only uniform data
Chebyshev’s inequality holds for any distribution with finite mean and variance. It guarantees at least 1−1/k² of values lie within k standard deviations of mean.
For k = 2, Chebyshev guarantees at least
A 50% within 2 SD
B 95% within 2 SD
C 25% within 2 SD
D 75% within 2 SD
Chebyshev: at least 1−1/k² within k SD. For k=2, 1−1/4 = 3/4 = 75%. This is a minimum guarantee for any distribution.
Which measure uses squared deviations and then averages
A Mean deviation
B Range
C Variance
D IQR
Variance is computed by squaring deviations from mean, summing them, and dividing by n (population) or n−1 (sample). Squaring emphasizes larger deviations.
In a frequency table, relative frequency is
A N/f
B f/N
C f×N
D f−N
Relative frequency is the proportion of observations in a class: class frequency divided by total frequency. It can be shown as a fraction, decimal, or percentage.
Cumulative relative frequency is obtained by
A Squaring frequencies
B Subtracting frequencies
C Adding relative frequencies
D Averaging midpoints
Cumulative relative frequency is the running total of relative frequencies. It increases from 0 to 1 (or 0% to 100%) and is useful for percentile interpretation.
If median is closer to Q1 than Q3, the data are
A Left skewed
B Symmetric
C Bimodal
D Right skewed
In right skew, upper tail is longer, so the upper half is more spread out. Median tends to lie closer to Q1 than Q3 because Q3 is pulled farther right.
If median is closer to Q3 than Q1, the data are
A Left skewed
B Right skewed
C Symmetric
D Uniform
In left skew, lower tail is longer, making the lower half more spread out. Median then tends to be closer to Q3 than Q1, reflecting longer spread on left.
A good summary for “spread around mean” is
A Mode
B Median
C Standard deviation
D Pie chart
SD measures typical dispersion around the mean in original units. It is widely used in statistics, probability, and data analysis due to its strong mathematical properties.
When comparing two datasets, equal IQR suggests
A Same mean
B Same range
C Same mode
D Same middle spread
IQR measures spread of the middle 50% data. If two datasets have the same IQR, their central spread is similar, even if means or extremes differ.