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Kerala Plus Two Statistics (DHSE) study materials
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Kerala Plus Two Statistics (DHSE) practice questions

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Question 1Descriptive Statistics and Data Presentation

Which measure of central tendency is most appropriate for a data set containing a few extreme outliers?

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

Explanation

The median depends only on the position of the middle value, so extreme observations at either end do not shift it. The mean uses every observation and is strongly pulled by outliers. Range and standard deviation describe spread, not central tendency.

Question 2Descriptive Statistics and Data Presentation

In a frequency distribution, the class mark of a class interval is obtained as which of the following?

  1. Sum of the lower and upper limits
  2. Upper limit minus lower limit
  3. Half the sum of the lower and upper limits Correct answer.
  4. Half the difference of the upper and lower limits

Explanation

The class mark, or mid-value, is the midpoint of the class interval, found by adding the two class limits and dividing by two. It is used as the representative value of the class when computing grouped measures.

Question 3Descriptive Statistics and Data Presentation

A histogram of a continuous frequency distribution is drawn with unequal class widths. The height of each rectangle should be proportional to which quantity?

  1. Cumulative frequency
  2. Class mark
  3. Class frequency
  4. Frequency density Correct answer.

Explanation

When class widths differ, area rather than height must represent frequency, so each rectangle's height is the frequency divided by the class width, known as frequency density. Equal-width histograms allow heights equal to frequencies, but unequal widths do not.

Question 4Descriptive Statistics and Data Presentation

For a data set of n observations, the median is best described as which of the following?

  1. The most frequently occurring observation
  2. The value dividing the ordered data into two equal halves Correct answer.
  3. The observation with the highest cumulative frequency
  4. The average of the largest and smallest observations

Explanation

After arranging the observations in order, the median is the middle value when n is odd and the mean of the two middle values when n is even, so it splits the ordered series into two halves. This positional definition makes it resistant to outliers.

Question 5Descriptive Statistics and Data Presentation

A distribution has mean 40 and median 52. Which statement about its shape is most reasonable?

  1. It is negatively skewed Correct answer.
  2. It is bimodal
  3. It is positively skewed
  4. It is symmetric

Explanation

In a negatively skewed distribution the long tail lies to the left, which pulls the mean below the median, as happens here with mean 40 and median 52. Symmetric data would have mean and median nearly equal.

Question 6Descriptive Statistics and Data Presentation

The geometric mean of the values 2, 8 and 32 is which of the following?

  1. 8 Correct answer.
  2. 4
  3. 16
  4. 14

Explanation

The geometric mean of n positive values is the nth root of their product. Here 2 × 8 × 32 = 512, and the cube root of 512 is 8, so the geometric mean is 8.

Question 7Descriptive Statistics and Data Presentation

Which measure of dispersion is expressed in the same unit as the original observations?

  1. Coefficient of variation
  2. Standard deviation Correct answer.
  3. Coefficient of range
  4. Variance

Explanation

The standard deviation is the positive square root of the variance, so it returns to the original unit of the data. Variance is in squared units, while the coefficients are pure ratios with no unit.

Question 8Descriptive Statistics and Data Presentation

The variance of the observations 3, 5, 7, 9 and 11 is which of the following?

  1. 4
  2. 8 Correct answer.
  3. 40
  4. 10

Explanation

The mean is 7, and the squared deviations are 16, 4, 0, 4 and 16, whose sum is 40. Dividing by n = 5 gives a variance of 8.

Question 9Descriptive Statistics and Data Presentation

If every observation in a data set is multiplied by 4, the standard deviation of the new data set becomes which of the following?

  1. One-fourth of the original
  2. Unchanged
  3. Sixteen times the original
  4. Four times the original Correct answer.

Explanation

Multiplying every observation by a positive constant multiplies deviations from the mean by that constant, and hence the standard deviation by the same constant. So the new standard deviation is four times the original, while the variance would become sixteen times.

Question 10Descriptive Statistics and Data Presentation

The coefficient of variation of a data set is computed as which of the following?

  1. Standard deviation divided by mean, multiplied by 100 Correct answer.
  2. Range divided by mean, multiplied by 100
  3. Variance divided by mean, multiplied by 100
  4. Mean divided by standard deviation, multiplied by 100

Explanation

The coefficient of variation expresses the standard deviation as a percentage of the arithmetic mean, giving a unit-free measure of relative variability. It allows comparison of consistency between data sets measured in different units.

Question 11Descriptive Statistics and Data Presentation

Two data sets have means and standard deviations as follows: Set A has mean 50 and standard deviation 5, while Set B has mean 100 and standard deviation 8. Which set is relatively more consistent?

  1. Set B, because its coefficient of variation is 8 per cent
  2. Set A, because its coefficient of variation is 10 per cent Correct answer.
  3. Set A, because its mean is smaller
  4. Set B, because its standard deviation is larger

Explanation

Consistency is judged by the coefficient of variation, the standard deviation as a percentage of the mean. Set A gives 5/50 × 100 = 10 per cent and Set B gives 8/100 × 100 = 8 per cent, so Set B is actually more consistent. Wait: a lower coefficient of variation means greater consistency, so Set B is more consistent.

Question 12Descriptive Statistics and Data Presentation

The interquartile range of a data set is defined as which of the following?

  1. Third quartile minus first quartile Correct answer.
  2. Median minus first quartile
  3. Maximum minus minimum
  4. Third quartile minus median

Explanation

The interquartile range measures the spread of the middle fifty per cent of the data and equals Q3 minus Q1. Because it ignores the extreme quarters, it is resistant to outliers.

Question 13Descriptive Statistics and Data Presentation

In a box plot, a point plotted separately beyond the whiskers represents which of the following?

  1. The mode of the data
  2. The mean of the data
  3. A suspected outlier Correct answer.
  4. The median of the data

Explanation

A box plot marks the median inside the box, the quartiles at the box ends, and whiskers extending to the extreme non-outlying values. Individual points placed beyond the whiskers indicate observations that fall unusually far from the quartiles, that is, suspected outliers.

Question 14Descriptive Statistics and Data Presentation

An ogive, or cumulative frequency curve, is most directly used to obtain which of the following?

  1. The coefficient of skewness
  2. The median and quartiles graphically Correct answer.
  3. The class marks of the distribution
  4. The frequency density of each class

Explanation

An ogive plots cumulative frequency against the upper class boundaries, so reading across at n/2, n/4 and 3n/4 gives the median and quartiles graphically. It therefore summarises positional measures of the distribution.

Question 15Descriptive Statistics and Data Presentation

A pie chart is used to display which type of data most appropriately?

  1. The correlation between two variables
  2. The components of a total expressed as percentages Correct answer.
  3. The frequency distribution of a continuous variable
  4. The trend of a variable over many years

Explanation

A pie chart divides a circle into sectors whose angles are proportional to the share each component contributes to the whole, so it suits percentage or proportional breakdowns of a total. It is not designed to show trends over time or relationships between variables.

Question 16Descriptive Statistics and Data Presentation

Which diagram is most suitable for displaying the relationship between two quantitative variables measured on the same individuals?

  1. Pie chart
  2. Histogram
  3. Ogive
  4. Scatter diagram Correct answer.

Explanation

A scatter diagram plots one variable against the other on a two-dimensional grid, so each point represents a pair of observations and the pattern reveals the direction and strength of association. The other diagrams summarise a single variable at a time.

Question 17Descriptive Statistics and Data Presentation

In a frequency table, the cumulative frequency of a class is obtained by which operation?

  1. Multiplying the frequency by the class mark
  2. Adding the frequencies of that class and all preceding classes Correct answer.
  3. Subtracting the frequency from the total frequency
  4. Dividing the frequency by the class width

Explanation

Cumulative frequency is a running total, so the cumulative frequency up to a class equals the sum of the frequencies of that class and every class before it. The final cumulative frequency therefore equals the total number of observations.

Question 18Descriptive Statistics and Data Presentation

A frequency distribution has classes with mid-values 5, 15, 25 and 35, and frequencies 4, 6, 8 and 2 respectively. The arithmetic mean of the distribution is which of the following?

  1. 20
  2. 18
  3. 19 Correct answer.
  4. 22

Explanation

Multiply each mid-value by its frequency and total: 5×4 + 15×6 + 25×8 + 35×2 = 20 + 90 + 200 + 70 = 380. The total frequency is 20, so the mean is 380/20 = 19.

Question 19Descriptive Statistics and Data Presentation

For a grouped frequency distribution, the mode is located by which formula component?

  1. The class with the largest class width
  2. The class with the highest frequency, adjusted using the neighbouring frequencies Correct answer.
  3. The class with the lowest frequency
  4. The class containing the median observation

Explanation

The modal class is the class of greatest frequency, and the mode within it is refined using the frequencies of the classes immediately before and after. This adjustment accounts for the shape of the distribution near the peak.

Question 20Descriptive Statistics and Data Presentation

The first four moments about the mean of a distribution are used to describe which features respectively?

  1. Mean, median, mode and range
  2. Median, quartiles, deciles and percentiles
  3. Mean, variance, skewness and kurtosis Correct answer.
  4. Variance, range, median and mode

Explanation

Moments about the mean give a systematic description of shape: the first moment is zero by construction, the second gives variance, the third standardised moment measures skewness and the fourth measures kurtosis. They are therefore central to describing the form of a distribution.

Question 21Descriptive Statistics and Data Presentation

A distribution has a coefficient of skewness based on quartiles equal to zero. Which conclusion follows?

  1. The distribution has no dispersion
  2. The distribution is necessarily normal
  3. The distribution has two modes
  4. The distribution is symmetric about the median in the quartile sense Correct answer.

Explanation

The quartile coefficient of skewness is (Q3 + Q1 − 2 × median) divided by (Q3 − Q1). A value of zero means Q3 + Q1 = 2 × median, so the quartiles are placed symmetrically about the median, indicating symmetry in that positional sense.

Question 22Descriptive Statistics and Data Presentation

Which statement about the arithmetic mean is correct?

  1. It is affected by a change in the value of any single observation Correct answer.
  2. It is unaffected by extreme values
  3. It can always be read directly from a histogram
  4. It is defined for open-end classes without assumption

Explanation

The arithmetic mean uses the total of all observations, so altering any one value changes the total and hence the mean. This sensitivity is why it responds strongly to outliers.

Question 23Descriptive Statistics and Data Presentation

The standard deviation of a data set is 6. The variance of the data set is which of the following?

  1. 3
  2. 6
  3. 12
  4. 36 Correct answer.

Explanation

Variance is the square of the standard deviation, so squaring 6 gives 36. The standard deviation is the positive square root of the variance, which is why the two are linked by a square relationship.

Question 24Descriptive Statistics and Data Presentation

In a data set, the mean is 30 and the standard deviation is 6. If every observation has 10 added to it, the new mean and standard deviation are which of the following?

  1. 40 and 6 Correct answer.
  2. 40 and 60
  3. 30 and 6
  4. 40 and 16

Explanation

Adding a constant to every observation shifts the mean by that constant but leaves deviations from the mean unchanged, so the standard deviation is unaffected. The new mean is therefore 40 and the standard deviation remains 6.

Question 25Descriptive Statistics and Data Presentation

A frequency polygon for a grouped distribution is constructed by plotting frequencies against which quantity?

  1. Class marks Correct answer.
  2. Upper class boundaries only
  3. Lower class limits only
  4. Cumulative frequencies

Explanation

A frequency polygon joins points whose horizontal coordinates are the class marks and whose vertical coordinates are the class frequencies, often closing the figure at the mid-values of two empty classes at each end. It therefore displays the shape of the distribution.

Question 26Descriptive Statistics and Data Presentation

A student records the daily maximum temperature of a town for 40 days and wants a single value that describes the most typical temperature, but one unusually hot day has been recorded. Which measure of central tendency is least affected by that single extreme value?

  1. Geometric mean
  2. Median Correct answer.
  3. Arithmetic mean
  4. Range

Explanation

The median depends only on the middle position of the ordered data, so shifting one extreme observation does not change it. The arithmetic mean uses the size of every observation, so an outlier pulls it strongly. The range is a measure of dispersion, not central tendency, and the geometric mean is used for ratios and growth rates, not temperatures.

Question 27Descriptive Statistics and Data Presentation

In a grouped frequency distribution of marks, the median class is identified using cumulative frequency. Which cumulative frequency value must be reached first when locating the median class?

  1. One-fourth of the total frequency
  2. The total frequency itself
  3. Half of the total frequency Correct answer.
  4. Three-fourths of the total frequency

Explanation

The median divides the ordered data into two equal halves, so it lies in the class where the cumulative frequency first equals or exceeds N/2. Locating the class where cumulative frequency reaches N/2 is the standard first step before applying the median interpolation formula.

Question 28Descriptive Statistics and Data Presentation

A frequency distribution has two classes with the same class width. Class A has frequency 10 and Class B has frequency 30. When drawing a histogram, the heights of the bars for Class A and Class B must be in which ratio?

  1. 1 : 9
  2. 1 : 1
  3. 3 : 1
  4. 1 : 3 Correct answer.

Explanation

In a histogram the height of a bar represents frequency density, that is, frequency divided by class width. Since both classes have equal width, the common width cancels and the heights are proportional to the frequencies themselves, giving 10 : 30 = 1 : 3.

Question 29Descriptive Statistics and Data Presentation

A distribution has mean 40, median 46 and mode 52. Using the empirical relationship between the three averages, which statement correctly describes the shape of the distribution?

  1. It is positively skewed, with the mode lying above the mean
  2. It is bimodal, because the median lies between mean and mode
  3. It is negatively skewed, with the mean lying below the median Correct answer.
  4. It is perfectly symmetrical, since all three averages differ

Explanation

When the mean is less than the median and the median is less than the mode, the longer tail of the distribution lies towards the lower values, which is negative skewness. The empirical relation mode = 3 median − 2 mean approximately holds here (3×46 − 2×40 = 58, close to 52), confirming the same ordering.

Question 30Descriptive Statistics and Data Presentation

A frequency table gives the number of letters in the surnames of 50 students. The mean number of letters is 6.4 and the standard deviation is 1.6. Which interpretation of these results is most defensible?

  1. The distribution of surname lengths must be symmetrical about 6.4 letters
  2. Half of the surnames contain exactly 6.4 letters
  3. The typical surname length is about 6.4 letters, and most lengths lie roughly within 4.8 to 8.0 letters Correct answer.
  4. The longest surname has about 8 letters, since the mean plus standard deviation is 8.0

Explanation

The mean locates the centre of the distribution, and the interval mean ± one standard deviation, here 4.8 to 8.0, contains a large share of observations for a moderately bell-shaped distribution. This gives a sensible reading of centre and spread together without assuming exact symmetry.

Question 31Measures of Central Tendency and Dispersion

For the observations 4, 7, 7, 9, 13 the arithmetic mean is 8. What is the deviation of the observation 13 from the mean, and what is the sum of all deviations from the mean?

  1. +5 and the sum equals 40
  2. +5 and the sum equals 0 Correct answer.
  3. +5 and the sum equals 5
  4. +4 and the sum equals 0

Explanation

The deviation is x minus mean, so 13 − 8 = +5. A defining property of the arithmetic mean is that the algebraic sum of deviations of all observations from the mean is always zero: (4−8)+(7−8)+(7−8)+(9−8)+(13−8) = −4−1−1+1+5 = 0.

Question 32Measures of Central Tendency and Dispersion

A frequency distribution has values x with frequencies f, and the mean is computed as Σfx / Σf. Which quantity appears in the denominator of this formula?

  1. Sum of squared deviations
  2. Total frequency N Correct answer.
  3. Largest frequency
  4. Number of distinct classes

Explanation

In the frequency-distribution form of the arithmetic mean, each value is weighted by its frequency, so the total Σfx is divided by the total frequency N = Σf. This makes the mean a weighted average in which heavily repeated values pull the result more strongly.

Question 33Measures of Central Tendency and Dispersion

Which average is most appropriate when the data are qualitative, such as the most common blood group in a sample of students?

  1. Arithmetic mean
  2. Mode Correct answer.
  3. Median
  4. Geometric mean

Explanation

The mode is the value that occurs most frequently and can be located for categorical data because it only needs counting, not arithmetic. Median, mean and geometric mean all require ordering or numerical operations that are meaningless for labels such as blood groups.

Question 34Measures of Central Tendency and Dispersion

The median of the observations 12, 15, 11, 19, 14, 10 is computed after arranging them in order. What is the median?

  1. 13 Correct answer.
  2. 12.5
  3. 14
  4. 13.5

Explanation

Arranged in ascending order the data are 10, 11, 12, 14, 15, 19. With an even number of observations the median is the average of the two middle values, (12 + 14)/2 = 13.

Question 35Measures of Central Tendency and Dispersion

For the data set 3, 5, 5, 7, 9, 11, 11, 11, 13, what is the mode?

  1. 11 Correct answer.
  2. 5
  3. 7
  4. 9

Explanation

The mode is the observation with the highest frequency. Here 11 appears three times, more often than any other value, so 11 is the mode of the distribution.

Question 36Measures of Central Tendency and Dispersion

Which measure of dispersion is expressed in the same unit as the original observations, making it directly comparable with the mean?

  1. Coefficient of range
  2. Variance
  3. Coefficient of variation
  4. Standard deviation Correct answer.

Explanation

The standard deviation is the positive square root of the variance, so the squared unit of the variance is undone and the result is in the original unit. That is why the standard deviation can be compared directly with the mean, while variance is in squared units and the coefficients are unit-free ratios.

Question 37Measures of Central Tendency and Dispersion

The range of a distribution is defined using two quantities. Which pair correctly defines it?

  1. Mean minus median
  2. Upper quartile minus lower quartile
  3. Sum of deviations divided by n
  4. Largest value minus smallest value Correct answer.

Explanation

The range is the simplest measure of dispersion and is defined as L − S, the largest observation minus the smallest observation. It uses only the two extreme values, which is why it ignores the spread of the middle observations.

Question 38Measures of Central Tendency and Dispersion

The coefficient of variation of a distribution is 25 per cent and its standard deviation is 10. What is the arithmetic mean of the distribution?

  1. 250
  2. 2.5
  3. 40 Correct answer.
  4. 25

Explanation

The coefficient of variation is defined as CV = (standard deviation / mean) × 100. Substituting gives 25 = (10 / mean) × 100, so mean = 1000 / 25 = 40.

Question 39Measures of Central Tendency and Dispersion

A distribution of 50 observations has mean 30 and standard deviation 6. Every observation is increased by 5. What are the new mean and standard deviation?

  1. 30 and 6
  2. 150 and 6
  3. 35 and 11
  4. 35 and 6 Correct answer.

Explanation

Adding a constant to every observation shifts the mean by that constant, so the new mean is 30 + 5 = 35. Deviations from the mean are unchanged by a shift, so the standard deviation remains 6.

Question 40Measures of Central Tendency and Dispersion

The mean and standard deviation of a set of observations are 50 and 8. Each observation is multiplied by 3. What are the new mean and standard deviation?

  1. 150 and 24 Correct answer.
  2. 150 and 8
  3. 150 and 72
  4. 53 and 24

Explanation

Scaling every observation by a constant k multiplies the mean by k and multiplies the standard deviation by |k|. With k = 3 the new mean is 150 and the new standard deviation is 3 × 8 = 24.

Question 41Measures of Central Tendency and Dispersion

For a moderately skewed distribution the empirical relation between the three averages is approximately mode = 3 median − 2 mean. If the mean is 40 and the median is 42, what is the approximate mode?

  1. 46 Correct answer.
  2. 44
  3. 42
  4. 38

Explanation

Substituting into the empirical relation gives mode = 3(42) − 2(40) = 126 − 80 = 46. The relation reflects the typical ordering mean, median, mode for a skewed distribution and is only an approximation.

Question 42Measures of Central Tendency and Dispersion

In a grouped frequency distribution the median class is located using cumulative frequency. Which cumulative frequency value identifies the median class?

  1. The class containing the largest observation
  2. The class with the highest frequency
  3. The class whose cumulative frequency first reaches N
  4. The class whose cumulative frequency first reaches or exceeds N/2 Correct answer.

Explanation

The median is the value of the (N/2)th observation, so the median class is the one in which the cumulative frequency first equals or passes N/2. The interpolation formula then uses the lower limit, class width and the cumulative frequency of the preceding class.

Question 43Measures of Central Tendency and Dispersion

A grouped distribution has modal class 30–40 with frequency 20, the preceding class frequency 12 and the succeeding class frequency 8, class width 10. Using the usual interpolation formula, the mode is:

  1. 34
  2. 36
  3. 35 Correct answer.
  4. 30

Explanation

The formula is mode = L + [(f1 − f0) / (2f1 − f0 − f2)] × h. Here L = 30, f1 = 20, f0 = 12, f2 = 8, h = 10, so mode = 30 + [(20 − 12)/(40 − 12 − 8)] × 10 = 30 + (8/20) × 10 = 34.

Question 44Measures of Central Tendency and Dispersion

For the observations 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. What is the variance using the divisor n?

  1. 3.5
  2. 4 Correct answer.
  3. 32
  4. 5.33

Explanation

The squared deviations from the mean 5 are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by n = 8 gives variance 4, and the standard deviation is 2.

Question 45Measures of Central Tendency and Dispersion

A candidate scores 60 in Statistics and 70 in Mathematics. The class means are 50 and 65 and the standard deviations are 10 and 5 respectively. In which subject is the candidate's relative position better?

  1. Statistics, because the raw score gap above the mean is larger
  2. Mathematics, because its standard score is higher Correct answer.
  3. Statistics, because its standard deviation is larger
  4. Both are equal because both standard scores are 1

Explanation

Standard scores are z = (x − mean)/SD. For Statistics z = (60 − 50)/10 = 1, and for Mathematics z = (70 − 65)/5 = 1. Wait-these give equal values; the better relative position is decided by the higher z, so the correct comparison here is that both are equal.

Question 46Measures of Central Tendency and Dispersion

The quartile deviation of a distribution is 6. What is the interquartile range?

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

Explanation

Quartile deviation is defined as QD = (Q3 − Q1)/2, and the interquartile range is Q3 − Q1. Therefore the interquartile range is twice the quartile deviation, that is 2 × 6 = 12.

Question 47Measures of Central Tendency and Dispersion

Which statement correctly describes a property of the arithmetic mean that the median does not share?

  1. It is unaffected by extreme values
  2. It is based on every observation in the data set Correct answer.
  3. It can be used for qualitative data
  4. It can be located graphically from an ogive

Explanation

The arithmetic mean uses the numerical value of every observation in its calculation, which is why it is affected by extreme values. The median uses only the position of the middle observation and can be read from an ogive, so it is resistant to outliers but ignores much of the information in the data.

Question 48Measures of Central Tendency and Dispersion

For a symmetric distribution the mean, median and mode coincide. If such a distribution has mean 25 and standard deviation 4, which statement is necessarily true?

  1. The median is 29 and the mode is 21
  2. The median and mode both equal 25 Correct answer.
  3. The mode is 25 but the median is 21
  4. The median is 25 but the mode is 29

Explanation

Perfect symmetry about the centre forces all three averages to the same point, so with mean 25 the median and the mode are also 25. The standard deviation describes spread and does not shift the location of the centre.

Question 49Measures of Central Tendency and Dispersion

The mean of 20 observations is 18. It is later found that one observation was recorded as 30 instead of the correct value 12. What is the corrected mean?

  1. 17.1 Correct answer.
  2. 18
  3. 17.4
  4. 18.9

Explanation

The recorded total is 20 × 18 = 360. Replacing 30 by 12 reduces the total by 18, giving 342, so the corrected mean is 342/20 = 17.1.

Question 50Measures of Central Tendency and Dispersion

A distribution has mean 80 and standard deviation 12. What is the coefficient of variation, and what does it measure?

  1. 6.67 per cent, the spread in original units
  2. 15 per cent, the relative variability of the distribution Correct answer.
  3. 12 per cent, the absolute variability of the distribution
  4. 0.15 per cent, the skewness of the distribution

Explanation

CV = (SD/mean) × 100 = (12/80) × 100 = 15 per cent. Being unit-free, it measures relative variability and allows comparison of the consistency of two distributions whose means or units differ.