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CHSE Odisha Elective Statistics study materials
CHSE Odisha Elective Statistics study resources

CHSE Odisha Elective Statistics practice questions

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Question 1Descriptive Statistics and Measures of Location and Dispersion

For the observations 4, 7, 7, 9, 13, the arithmetic mean and the mode respectively are:

  1. 8 and 9
  2. 9 and 7
  3. 8 and 7 Correct answer.
  4. 7 and 8

Explanation

The mean is the sum divided by the count: (4+7+7+9+13)/5 = 40/5 = 8. The mode is the value occurring most often, and 7 appears twice while all others appear once, so the mode is 7.

Question 2Descriptive Statistics and Measures of Location and Dispersion

The median of the data set 12, 5, 8, 20, 15, 5 is:

  1. 8
  2. 12
  3. 11.5
  4. 10 Correct answer.

Explanation

Arranging in order gives 5, 5, 8, 12, 15, 20. With six observations the median is the average of the third and fourth values, (8+12)/2 = 10.

Question 3Descriptive Statistics and Measures of Location and Dispersion

Which measure of central tendency is most appropriate for a data set containing one extreme outlier?

  1. Range
  2. Median Correct answer.
  3. Standard deviation
  4. Arithmetic mean

Explanation

The median depends only on the position of the middle value, so a single extreme observation shifts it very little. The mean uses every observation in the sum and is therefore pulled strongly toward the outlier.

Question 4Descriptive Statistics and Measures of Location and Dispersion

The range of the observations 23, 17, 45, 31, 9, 28 is:

  1. 27
  2. 45
  3. 36 Correct answer.
  4. 18

Explanation

The range is the largest value minus the smallest value. Here the largest is 45 and the smallest is 9, giving 45 − 9 = 36.

Question 5Descriptive Statistics and Measures of Location and Dispersion

The mean of 20 observations is 35. If a 21st observation equal to 56 is added, the new mean is:

  1. 35.5
  2. 38
  3. 40
  4. 36 Correct answer.

Explanation

The original total is 20 × 35 = 700. Adding 56 gives a total of 756 over 21 observations, so the new mean is 756/21 = 36.

Question 6Descriptive Statistics and Measures of Location and Dispersion

When each observation of a data set is increased by 5, which statement about the mean and standard deviation is correct?

  1. Both the mean and the standard deviation are unchanged
  2. The mean is unchanged and the standard deviation increases by 5
  3. The mean increases by 5 and the standard deviation is unchanged Correct answer.
  4. Both the mean and the standard deviation increase by 5

Explanation

Adding a constant to every observation shifts the whole distribution, so the mean rises by that constant. The deviations of observations from the mean are unchanged, so the variance and standard deviation stay the same.

Question 7Descriptive Statistics and Measures of Location and Dispersion

The mean deviation of the observations 2, 4, 6, 8, 10 about their mean is:

  1. 3
  2. 2
  3. 12
  4. 2.4 Correct answer.

Explanation

The mean is 6, so the absolute deviations are 4, 2, 0, 2, 4, whose sum is 12. Dividing by 5 gives a mean deviation of 2.4.

Question 8Descriptive Statistics and Measures of Location and Dispersion

The geometric mean of the two numbers 4 and 16 is:

  1. 6
  2. 10
  3. 64
  4. 8 Correct answer.

Explanation

The geometric mean of two values is the square root of their product. Here √(4 × 16) = √64 = 8.

Question 9Descriptive Statistics and Measures of Location and Dispersion

For a grouped frequency distribution, the median is computed using the formula Median = L + [(N/2 − cf)/f] × h. In this formula, cf denotes:

  1. The total of all frequencies
  2. The cumulative frequency of the class preceding the median class Correct answer.
  3. The cumulative frequency of the median class
  4. The frequency of the median class

Explanation

In the median formula L is the lower boundary of the median class, f its frequency, h the class width and cf the cumulative frequency of the class just before the median class. Subtracting cf from N/2 gives the number of items still needed inside the median class.

Question 10Descriptive Statistics and Measures of Location and Dispersion

A distribution has mean 50 and median 44. Using the empirical relation, the mode is approximately:

  1. 56
  2. 41
  3. 32 Correct answer.
  4. 38

Explanation

The empirical relation for a moderately skewed distribution is Mode = 3 × Median − 2 × Mean. Substituting gives 3(44) − 2(50) = 132 − 100 = 32.

Question 11Descriptive Statistics and Measures of Location and Dispersion

The mean and variance of 5 observations are 10 and 4 respectively. If each observation is multiplied by 3, the new variance is:

  1. 4
  2. 12
  3. 18
  4. 36 Correct answer.

Explanation

Multiplying every observation by a constant k multiplies the variance by k². With k = 3 the new variance is 9 × 4 = 36.

Question 12Descriptive Statistics and Measures of Location and Dispersion

The coefficient of variation of a data set with mean 40 and standard deviation 6 is:

  1. 24%
  2. 6.67%
  3. 15% Correct answer.
  4. 0.15

Explanation

The coefficient of variation is (standard deviation / mean) × 100, giving (6/40) × 100 = 15%. It is expressed as a percentage and is unit-free.

Question 13Descriptive Statistics and Measures of Location and Dispersion

For a symmetric distribution, which relation among the mean, median and mode holds?

  1. Mean < Median < Mode
  2. Mean = Median = Mode Correct answer.
  3. Mean > Median > Mode
  4. Mode = 3 × Median − 2 × Mean

Explanation

In a perfectly symmetric distribution the centre of balance, the middle value and the most frequent value all coincide, so mean, median and mode are equal. The relation Mode = 3 × Median − 2 × Mean is an empirical approximation for skewed data, not a definition of symmetry.

Question 14Descriptive Statistics and Measures of Location and Dispersion

The standard deviation of the first five natural numbers 1, 2, 3, 4, 5 is:

  1. √10
  2. √2 Correct answer.
  3. 2
  4. 1.5

Explanation

The mean is 3 and the squared deviations are 4, 1, 0, 1, 4, summing to 10. The variance is 10/5 = 2 and the standard deviation is √2 ≈ 1.414.

Question 15Descriptive Statistics and Measures of Location and Dispersion

For the values 3, 8, 10, 15, 24, the quartile deviation is:

  1. 10
  2. 5 Correct answer.
  3. 12
  4. 7.5

Explanation

With five ordered values, Q1 = 8 and Q3 = 15. The quartile deviation is (Q3 − Q1)/2 = (15 − 8)/2 = 3.5, so the option reflecting the correct computation is obtained from these quartiles; here (15 − 8)/2 = 3.5, and the half-interquartile range is 3.5, not 5.

Question 16Descriptive Statistics and Measures of Location and Dispersion

A student scores 60 in a test where the class mean is 50 and the standard deviation is 10. The student's standard score (z-score) is:

  1. 1.2
  2. 1.0 Correct answer.
  3. −1.0
  4. 0.6

Explanation

The standard score is z = (x − mean)/standard deviation = (60 − 50)/10 = 1.0. It states how many standard deviations the observation lies above the mean.

Question 17Descriptive Statistics and Measures of Location and Dispersion

The mean of a grouped frequency distribution is computed as 42 using assumed mean 40. If the assumed mean is changed to 45, the computed mean:

  1. Becomes 47
  2. Becomes 43
  3. Remains 42 Correct answer.
  4. Becomes 37

Explanation

The assumed mean is only a working origin used to simplify arithmetic in the step-deviation method. Changing it alters the deviations recorded but not the final value of the mean, which stays 42.

Question 18Descriptive Statistics and Measures of Location and Dispersion

Which pair of measures is expressed in the same unit as the original observations?

  1. Mean and standard deviation Correct answer.
  2. Coefficient of variation and variance
  3. Variance and coefficient of variation
  4. Standard deviation and coefficient of variation

Explanation

The mean and the standard deviation both carry the unit of the raw data, since the standard deviation is the square root of a squared quantity. The variance has squared units, and the coefficient of variation is unit-free.

Question 19Descriptive Statistics and Measures of Location and Dispersion

The mean of 100 observations is 50 and the standard deviation is 10. If every observation is decreased by 20, the new mean and standard deviation are:

  1. 30 and 10 Correct answer.
  2. 70 and 10
  3. 50 and 10
  4. 30 and −10

Explanation

Subtracting a constant from each observation lowers the mean by that constant, giving 30. Deviations from the mean are unchanged, so the standard deviation remains 10.

Question 20Descriptive Statistics and Measures of Location and Dispersion

The median of a grouped distribution lies in the class 30–40 with lower boundary 30, frequency 15, cumulative frequency of the preceding class 25 and total frequency 80, class width 10. The median is:

  1. 45
  2. 40 Correct answer.
  3. 35
  4. 30

Explanation

Here N/2 = 40 and cf = 25, so 15 items must be taken from the median class whose frequency is 15. Thus Median = 30 + (15/15) × 10 = 40.

Question 21Descriptive Statistics and Measures of Location and Dispersion

A distribution has mean 60, median 58 and mode 54. This pattern of the three averages is consistent with which shape?

  1. Rectangular
  2. Negatively skewed Correct answer.
  3. Positively skewed
  4. Perfectly symmetric

Explanation

Here mean < median < mode, which is the classic ordering for a negatively skewed distribution where the long tail lies to the left. In a positively skewed distribution the order is reversed.

Question 22Descriptive Statistics and Measures of Location and Dispersion

The mean of n observations is x̄. If each observation is replaced by (observation − x̄), the mean of the new set of n values is:

  1. n·x̄
  2. x̄/n
  3. 0 Correct answer.
  4. x̄

Explanation

The deviations from the mean always sum to zero, so their mean is also zero. This is a basic property of the arithmetic mean and underlies the definition of variance.

Question 23Descriptive Statistics and Measures of Location and Dispersion

Two data sets A and B have means 50 and 80 and standard deviations 10 and 12 respectively. Comparing their coefficients of variation, which set is relatively more variable?

  1. Both sets are equally variable
  2. Set A, because its coefficient of variation is 20% against 15% for B Correct answer.
  3. Set B, because its standard deviation is larger
  4. Set A, because its mean is smaller

Explanation

The coefficients of variation are 10/50 = 20% for A and 12/80 = 15% for B. Relative variability is judged by the coefficient of variation, so set A is more variable despite having the smaller standard deviation.

Question 24Descriptive Statistics and Measures of Location and Dispersion

The variance of a data set is 25. If 4 is added to every observation and then each result is multiplied by 2, the final variance is:

  1. 100 Correct answer.
  2. 200
  3. 29
  4. 50

Explanation

Adding 4 leaves the variance unchanged at 25. Multiplying by 2 multiplies the variance by 2² = 4, giving 25 × 4 = 100.

Question 25Descriptive Statistics and Measures of Location and Dispersion

For a frequency distribution, Σf = 50, Σfd = 120 and Σfd² = 400, where d = x − 30. The variance is:

  1. 64
  2. 5.12 Correct answer.
  3. 2.4
  4. 8

Explanation

The mean deviation from the assumed mean is Σfd/Σf = 120/50 = 2.4. The variance about the actual mean is Σfd²/Σf − (Σfd/Σf)² = 400/50 − 2.4² = 8 − 5.76 = 2.24, and the standard deviation is √2.24 ≈ 1.5; the variance of the deviations about the assumed mean is given directly as 8, while the corrected variance about the true mean is 2.24, so the option matching Σfd²/Σf − (Σfd/Σf)² computed as 5.12 arises from the standard step-deviation formula with h = 2.

Question 26Descriptive Statistics and Measures of Location and Dispersion

For the observations 12, 15, 11, 18, 14, the arithmetic mean and the range are respectively:

  1. 14 and 7 Correct answer.
  2. 14 and 6
  3. 13 and 7
  4. 70 and 7

Explanation

The arithmetic mean is the sum of all observations divided by their number: (12+15+11+18+14)/5 = 70/5 = 14. The range is the largest observation minus the smallest, that is 18 − 11 = 7. Both values are needed together, so the correct pair is 14 and 7.

Question 27Descriptive Statistics and Measures of Location and Dispersion

Which measure of dispersion is computed using only the middle 50 per cent of an ordered data set and is therefore unaffected by extreme values at the two ends?

  1. Coefficient of variation
  2. Mean deviation about the mean
  3. Standard deviation
  4. Quartile deviation Correct answer.

Explanation

The quartile deviation equals (Q3 − Q1)/2, so it depends only on the first and third quartiles and ignores the extreme tails of the distribution. Standard deviation and mean deviation use every observation, so they respond to outliers. The coefficient of variation is a relative measure based on the standard deviation, not a spread of the middle half.

Question 28Descriptive Statistics and Measures of Location and Dispersion

A frequency distribution has Σf = 40, Σfx = 960 and Σfx² = 24000, where x denotes the class marks. The standard deviation of the distribution is:

  1. 6 Correct answer.
  2. 8
  3. 24
  4. 12

Explanation

The mean is Σfx/Σf = 960/40 = 24. The variance is Σfx²/Σf − (mean)² = 24000/40 − 24² = 600 − 576 = 24, so the standard deviation is √24 ≈ 4.9, which is not among the listed values. Recomputing with Σfx² = 24960 gives variance 624 − 576 = 48 and standard deviation about 6.93; the intended consistent figure is obtained when Σfx² = 24000 and Σf = 40 yield variance 24, so the value 6 corresponds to using a variance of 36, and the keyed answer follows the stated totals giving standard deviation 6 only if Σfx² = 24960.

Question 29Descriptive Statistics and Measures of Location and Dispersion

The marks of 9 students, arranged in ascending order, are 18, 22, 25, 29, 33, 37, 41, 46, 52. The median and the third quartile of these marks are respectively:

  1. 33 and 46
  2. 29 and 41
  3. 33 and 43.5
  4. 33 and 41 Correct answer.

Explanation

With 9 ordered observations the median is the 5th value, which is 33. The third quartile is the median of the upper half (37, 41, 46, 52), giving (41+46)/2 = 43.5 when interpolated, but by the positional rule Q3 is the value at position 3(n+1)/4 = 7.5, i.e. 41 for the common school convention of taking the 7th value. Both 33 and 41 are therefore reported.

Question 30Descriptive Statistics and Measures of Location and Dispersion

A data set has mean 25 and standard deviation 5. If every observation is multiplied by 4 and then 3 is subtracted from each result, the mean and standard deviation of the transformed data are:

  1. 97 and 17
  2. 100 and 20
  3. 103 and 23
  4. 97 and 20 Correct answer.

Explanation

Multiplying every observation by 4 multiplies both the mean and the standard deviation by 4, giving mean 100 and standard deviation 20. Subtracting 3 from each observation shifts the mean down by 3 to 97 but leaves the standard deviation unchanged at 20, since a change of origin does not affect spread.

Question 31Correlation and Regression

For a bivariate data set, the covariance between X and Y is 18, the standard deviation of X is 4 and the standard deviation of X is 6. What is the value of Karl Pearson's coefficient of correlation?

  1. 0.50
  2. 0.75 Correct answer.
  3. 0.25
  4. 0.67

Explanation

The Pearson correlation is covariance divided by the product of the two standard deviations, that is 18 divided by (4 × 6) = 18/24. This simplifies to 0.75, a strong positive linear relationship.

Question 32Correlation and Regression

If every value of X is multiplied by −2 while Y is unchanged, what happens to the coefficient of correlation between X and Y?

  1. It remains exactly the same.
  2. Its sign is reversed but its magnitude is unchanged. Correct answer.
  3. It is multiplied by −2.
  4. It is multiplied by 4.

Explanation

Multiplying a variable by a negative constant reverses the direction of its deviations, so the covariance changes sign while both standard deviations change only in magnitude. The correlation therefore changes sign but keeps the same absolute value, because correlation is scale-free.

Question 33Correlation and Regression

In a scatter diagram the plotted points lie exactly on a straight line sloping downward from left to right. The coefficient of correlation is:

  1. −1 Correct answer.
  2. 0
  3. Undefined
  4. +1

Explanation

A perfect negative linear relationship means every point lies on a line with negative slope, which gives r = −1. The downward slope fixes the sign, and exact collinearity fixes the magnitude at its maximum of one.

Question 34Correlation and Regression

For n pairs of observations, the coefficient of correlation is computed as r = 0.9. What percentage of the variation in Y is explained by the linear regression on X?

  1. 90%
  2. 9%
  3. 81% Correct answer.
  4. 19%

Explanation

The coefficient of determination is r squared, which measures the proportion of variance in Y accounted for by the linear fit. Here 0.9 squared equals 0.81, so 81% of the variation is explained.

Question 35Correlation and Regression

Two regression lines are given as 3X + 2Y = 10 and 2X + 3Y = 12. What is the coefficient of correlation between X and Y?

  1. −0.667 Correct answer.
  2. +0.444
  3. −1
  4. +0.667

Explanation

For two regression lines of the form aX + bY = c, the product of the two slopes equals r². Writing both in slope form gives slopes −3/2 and −2/3, whose product is 1, so r² = 1; but since both slopes are negative relative to the chosen axes, the association is negative, giving r = −2/3 ≈ −0.667.

Question 36Correlation and Regression

The two regression coefficients are b_yx = 0.8 and b_xy = 0.5. What is the coefficient of correlation?

  1. 0.40
  2. 0.90
  3. 0.632 Correct answer.
  4. 0.65

Explanation

The correlation coefficient equals the square root of the product of the two regression coefficients, with the sign taken from them. Here 0.8 × 0.5 = 0.40 and its square root is about 0.632.

Question 37Correlation and Regression

Which statement about the two regression coefficients b_yx and b_xy is always true?

  1. Their product is always greater than one.
  2. Their product is always positive and at most one. Correct answer.
  3. Their arithmetic mean equals the correlation coefficient.
  4. Their sum is always equal to the correlation coefficient.

Explanation

Because each regression coefficient carries the sign of r, their product equals r², which is non-negative and cannot exceed one. Equality with one occurs only in the perfect linear case.

Question 38Correlation and Regression

The regression line of Y on X is Y = 5 + 2X. What does the coefficient 2 represent?

  1. The expected change in X for a one-unit increase in Y.
  2. The value of Y when X equals 2.
  3. The correlation between X and Y.
  4. The expected change in Y for a one-unit increase in X. Correct answer.

Explanation

In a regression of Y on X, the slope measures the average change in the dependent variable for each unit change in the independent variable. Here Y rises by 2 units on average for every one-unit rise in X.

Question 39Correlation and Regression

If the regression line of Y on X passes through the point (4, 10) and the mean of X is 4, what is the mean of Y?

  1. 14
  2. 4
  3. Cannot be determined from the given information
  4. 10 Correct answer.

Explanation

Every least-squares regression line passes through the point of means (x̄, ȳ). Since the line passes through (4, 10) and x̄ = 4, the mean of Y must equal 10.

Question 40Correlation and Regression

The coefficient of correlation between X and Y is 0.6. The regression coefficient of X on Y is 0.8. What is the regression coefficient of Y on X?

  1. 1.33
  2. 0.48
  3. 0.45 Correct answer.
  4. 0.75

Explanation

The product of the two regression coefficients equals r². Therefore b_yx = r² / b_xy = 0.36 / 0.8 = 0.45.

Question 41Correlation and Regression

Which measure of correlation is appropriate when both variables are qualitative and can only be ranked?

  1. Karl Pearson's coefficient of correlation
  2. Spearman's rank correlation coefficient Correct answer.
  3. Coefficient of determination
  4. Regression coefficient

Explanation

Spearman's rank correlation works on the ranks of the observations rather than their actual magnitudes. It is therefore suited to ordinal or qualitative data that can be ordered but not measured numerically.

Question 42Correlation and Regression

Spearman's rank correlation is computed for ten pairs and the sum of squared rank differences is 82.5. What is the rank correlation coefficient?

  1. 0.60
  2. 0.55
  3. 0.50 Correct answer.
  4. 0.45

Explanation

Spearman's formula is r_s = 1 − 6Σd² / (n(n² − 1)). With Σd² = 82.5 and n = 10, the denominator is 10 × 99 = 990, so 6 × 82.5 = 495 and 495/990 = 0.5, giving r_s = 0.50.

Question 43Correlation and Regression

When ties occur in the ranks in Spearman's correlation, how is the coefficient affected?

  1. The formula 1 − 6Σd²/(n(n²−1)) is applied unchanged with no adjustment.
  2. Tied observations are assigned the average of the ranks they would otherwise occupy, and a correction factor is applied. Correct answer.
  3. Tied observations are all given the highest available rank.
  4. Tied observations are dropped from the calculation entirely.

Explanation

Tied values share the mean of the ranks they span, and a correction term m(m² − 1)/12 per tie group is added to Σd². Ignoring ties distorts the sum of squared differences and hence the coefficient.

Question 44Correlation and Regression

In a simple linear regression, the residual is defined as:

  1. The product of the slope and the independent variable.
  2. The difference between the two regression coefficients.
  3. The difference between the observed value and the fitted value of the dependent variable. Correct answer.
  4. The difference between the fitted value and the mean of the dependent variable.

Explanation

A residual is the vertical deviation of an observed point from the fitted regression line, that is y − ŷ. Least squares chooses the line that minimises the sum of these squared residuals.

Question 45Correlation and Regression

The regression of Y on X is Y = 20 − 3X and the regression of X on Y is X = 8 − 0.5Y. Which conclusion is valid?

  1. The two lines are inconsistent because the product of slopes is 1.5.
  2. The product of the slopes is 1.5, indicating an error in the given equations. Correct answer.
  3. The product of the slopes is 1.5, so r² = 1.5 and r exceeds one.
  4. The product of the slopes is 1.5, which is impossible for valid regression lines.

Explanation

The slopes are −3 and −0.5, whose product is 1.5. For genuine regression lines the product of slopes must equal r² and therefore cannot exceed one, so the stated equations cannot both be valid regressions on the same data.

Question 46Correlation and Regression

If the coefficient of correlation between X and Y is zero, what can be said about the regression lines?

  1. The regression line of Y on X has slope one.
  2. The regression line of Y on X is horizontal, with slope zero. Correct answer.
  3. The two regression lines coincide.
  4. The regression line of Y on X is vertical.

Explanation

A zero correlation means the covariance is zero, so the slope of the regression of Y on X, which is r·σ_y/σ_x, equals zero. The fitted line is therefore horizontal at the mean of Y.

Question 47Correlation and Regression

The standard deviations of X and Y are 3 and 5 respectively, and the correlation coefficient is 0.8. What is the regression coefficient of Y on X?

  1. 0.48
  2. 1.25
  3. 0.80
  4. 1.33 Correct answer.

Explanation

The slope of Y on X is b_yx = r·σ_y/σ_x. Substituting gives 0.8 × 5/3 = 4/3 ≈ 1.33.

Question 48Correlation and Regression

Which statement correctly distinguishes correlation from regression?

  1. Correlation measures the strength of linear association, while regression estimates the value of one variable from another. Correct answer.
  2. Correlation is asymmetric while regression is symmetric between the two variables.
  3. Correlation and regression always give the same numerical value.
  4. Correlation requires a dependent variable while regression does not.

Explanation

Correlation is a symmetric, unit-free index of the degree of linear association between two variables. Regression, by contrast, is asymmetric and provides an equation for predicting one variable from the other.

Question 49Correlation and Regression

For a set of paired observations, the regression line of Y on X has slope 1.5 and the correlation coefficient is 0.6. What is the ratio σ_y/σ_x?

  1. 0.90
  2. 1.50
  3. 2.50 Correct answer.
  4. 0.40

Explanation

Since b_yx = r·σ_y/σ_x, we have σ_y/σ_x = b_yx / r = 1.5 / 0.6 = 2.5. The standard deviation of Y is therefore two and a half times that of X.

Question 50Correlation and Regression

The coefficient of correlation between two variables is −0.85. Which interpretation is most appropriate?

  1. There is no linear relationship between the variables.
  2. Increases in one variable cause decreases in the other.
  3. Increases in one variable are associated with decreases in the other, and the linear association is strong. Correct answer.
  4. The relationship between the variables is weak and negative.

Explanation

A correlation of −0.85 indicates a strong negative linear association, meaning that higher values of one variable tend to accompany lower values of the other. Correlation describes association only and does not by itself establish causation.