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Karnataka II PUC Statistics (KSEAB) study materials
Karnataka II PUC Statistics (KSEAB) study resources

Karnataka II PUC Statistics (KSEAB) practice questions

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Question 1Statistics: Meaning, Scope and Data Collection

In statistics, a characteristic that can take different values in different units of a population, such as height or marks scored, is called a:

  1. Variable Correct answer.
  2. Census
  3. Parameter
  4. Constant

Explanation

A variable is a measurable characteristic whose value differs from unit to unit, so height or marks recorded for different students are observations of a variable. A constant has the same fixed value for every unit, while a parameter describes a population summary measure rather than the characteristic itself.

Question 2Statistics: Meaning, Scope and Data Collection

A researcher records the number of members in each household of a village. The variable 'number of members' is best classified as:

  1. Qualitative
  2. Discrete Correct answer.
  3. Nominal
  4. Continuous

Explanation

A discrete variable takes only distinct whole-number values obtained by counting, and household size is counted as 1, 2, 3 and so on with no intermediate values possible. Continuous variables arise from measurement and can take any value in an interval, which does not fit a count of persons.

Question 3Statistics: Meaning, Scope and Data Collection

Which of the following is an example of primary data?

  1. Rainfall statistics taken from a meteorological department website
  2. Census tables downloaded from the official census portal
  3. Marks obtained by students recorded by their own teacher during a class test Correct answer.
  4. Figures copied from a published state economic survey

Explanation

Primary data are collected first-hand by the investigator for the specific enquiry at hand, so marks recorded directly by the teacher who conducted the test are primary. Data already gathered and published by another agency are secondary, however reliable that agency may be.

Question 4Statistics: Meaning, Scope and Data Collection

The method in which the investigator personally contacts respondents and asks them questions face to face is known as:

  1. Mailed questionnaire method
  2. Indirect oral investigation
  3. Telephone interview method
  4. Personal interview method Correct answer.

Explanation

The personal interview method involves direct face-to-face contact between the investigator and the respondent, allowing clarification and observation of reactions. The other methods rely on written questionnaires, telephones or third parties rather than direct personal contact.

Question 5Statistics: Meaning, Scope and Data Collection

A pilot survey is conducted before the main investigation mainly to:

  1. Replace the need for a sampling frame
  2. Guarantee that non-sampling errors become zero
  3. Test the questionnaire and plan on a small scale and remove defects Correct answer.
  4. Increase the sample size to the whole population

Explanation

A pilot survey is a small-scale trial run that reveals ambiguous questions, unsuitable wording and practical difficulties so that the schedule and plan can be corrected before full-scale work. It is a pre-test of the instrument and procedure, not a means of enlarging the sample or eliminating error entirely.

Question 6Statistics: Meaning, Scope and Data Collection

Which measure of central tendency is most appropriate for summarising data measured on a nominal scale, such as blood group?

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

Explanation

Nominal data are merely labels without order or numerical magnitude, so the only meaningful average is the mode, the most frequently occurring category. Means and medians require numerical or at least ordered values, which nominal categories do not provide.

Question 7Statistics: Meaning, Scope and Data Collection

In a survey of 500 students, the investigator stands at the college gate and questions the first 50 students who arrive. The main limitation of this approach is that:

  1. It may produce a biased sample because selection is not random Correct answer.
  2. It is a complete enumeration of the population
  3. It always requires a larger budget than a census
  4. It gives every student a known and equal chance of selection

Explanation

Questioning whoever happens to arrive first is a convenience or judgement selection, so students with different arrival habits are systematically excluded and the sample may not represent the population. Probability sampling, by contrast, gives each unit a known chance of selection and supports valid inference.

Question 8Statistics: Meaning, Scope and Data Collection

A lottery draw assigns each of 1,000 employees a numbered slip, and 100 slips are drawn at random without replacement. This procedure is an example of:

  1. Quota sampling
  2. Simple random sampling Correct answer.
  3. Purposive sampling
  4. Snowball sampling

Explanation

Drawing numbered slips by lottery gives every employee an equal and known chance of selection, which is the defining feature of simple random sampling. The other methods select units by interviewer quotas, deliberate choice or referral, and none of them gives equal probability of selection.

Question 9Statistics: Meaning, Scope and Data Collection

Which of the following statements about a census is correct?

  1. It cannot be used to study rare characteristics
  2. It studies only a representative part of the population
  3. It is generally more economical than a sample survey
  4. It provides information on every unit of the population Correct answer.

Explanation

A census enumerates every unit of the population, so it yields complete information with no sampling error, although it is costly and time-consuming. A sample survey, by contrast, studies only a selected part of the population and is generally cheaper.

Question 10Statistics: Meaning, Scope and Data Collection

A questionnaire item reads: 'Do you agree that the college canteen should continue serving tasty, affordable food?' The main defect of this question is that it:

  1. Asks two issues at once and uses leading wording Correct answer.
  2. Uses an open-ended format requiring long answers
  3. Contains a double negative
  4. Is too short to be understood by respondents

Explanation

The item combines two separate issues, taste and affordability, and also pushes the respondent towards agreement by using the loaded words 'tasty' and 'affordable'. A good schedule question should be short, neutral, unambiguous and confined to a single point.

Question 11Statistics: Meaning, Scope and Data Collection

A firm wishes to estimate the proportion of defective bulbs in a consignment of 20,000 bulbs by destructive testing. Which approach is most suitable and why?

  1. Census, because the consignment is homogeneous
  2. Sample survey, because testing destroys the bulbs and the population is large Correct answer.
  3. Census, because complete accuracy is always required
  4. Sample survey, because sampling eliminates all errors

Explanation

Destructive testing makes a census impossible in practice, since testing every bulb would leave none to sell, and the large consignment makes a sample both adequate and economical. Sampling does not remove all error, but it permits valid inference about the whole consignment at a small cost.

Question 12Statistics: Meaning, Scope and Data Collection

A newspaper reports that 'the average monthly income of families in a district is Rs. 24,000', based on data compiled from income-tax returns. Which statement about this figure is most accurate?

  1. It is primary data collected by the reader and free of bias
  2. It is a discrete variable measured on a nominal scale
  3. It is secondary data and may understate income because many families do not file returns Correct answer.
  4. It is a population parameter computed from a complete census of all families

Explanation

The figure is compiled by another agency from tax records, so it is secondary data for the reader, and tax returns exclude non-filers and unreported income, which biases the average downward. Data quality in secondary sources must be judged by reliability, coverage and method of collection.

Question 13Classification, Tabulation and Presentation of Data

In the classification of data, which basis groups units according to measurable characteristics such as age, height or income?

  1. Qualitative classification
  2. Chronological classification
  3. Quantitative classification Correct answer.
  4. Spatial classification

Explanation

Quantitative classification uses numerical variables that can be measured and expressed in figures, so units are grouped by magnitude of a measurable characteristic. Qualitative classification instead groups by descriptive attributes, while spatial and chronological classification are forms of classification by time and place.

Question 14Classification, Tabulation and Presentation of Data

A frequency distribution in which the class intervals are not equal in width is called:

  1. An open-end distribution
  2. A cumulative frequency distribution
  3. An unequal class interval distribution Correct answer.
  4. A continuous distribution

Explanation

When the widths of successive class intervals differ, the distribution is described as having unequal or unequal-width class intervals, and frequencies must be adjusted before comparison. Equal-width distributions are usually preferred because raw frequencies are directly comparable across classes.

Question 15Classification, Tabulation and Presentation of Data

In a table, the row and column headings that explain the nature of the data presented are collectively known as:

  1. The captions or stubs Correct answer.
  2. The source note
  3. The footnote
  4. The title

Explanation

Captions are the column headings and stubs are the row headings; together they identify what each row and column of the body contains. The title states the overall subject of the table, while footnotes and source notes appear below the body.

Question 16Classification, Tabulation and Presentation of Data

Which of the following is a component of a well-labelled statistical table but is placed above the table body?

  1. Source note
  2. Table number and title Correct answer.
  3. Prefatory note
  4. Footnote

Explanation

The table number and title head the table and are placed above the body so the reader knows what the data represent before reading them. Footnotes and source notes conventionally appear below the table body.

Question 17Classification, Tabulation and Presentation of Data

A pie diagram is most appropriate for presenting:

  1. The trend of a variable over several decades
  2. The percentage composition of a total divided into components Correct answer.
  3. The cumulative frequency of a continuous variable
  4. The relationship between two quantitative variables

Explanation

A pie diagram divides a circle into sectors proportional to the share of each component in the total, so it shows percentage composition of a whole. Trend over time is better shown by a line or time-series graph, and relationship between two variables by a scatter diagram.

Question 18Classification, Tabulation and Presentation of Data

Which graph is drawn using cumulative frequencies and is used to locate the median and quartiles graphically?

  1. Pictogram
  2. Frequency polygon
  3. Histogram
  4. Ogive Correct answer.

Explanation

An ogive is the graph of a cumulative frequency distribution, plotted against class boundaries, and partition values such as the median and quartiles are read from it. A histogram shows the original class frequencies as adjacent rectangles.

Question 19Classification, Tabulation and Presentation of Data

The number of students in a college by course and by sex is to be presented in a single table. The most suitable table is:

  1. A frequency distribution table
  2. A two-way table Correct answer.
  3. A simple table
  4. A time-series table

Explanation

Two characteristics, course and sex, are cross-classified, so a two-way table with courses as rows and sex as columns presents the data compactly. A simple table handles only one characteristic at a time.

Question 20Classification, Tabulation and Presentation of Data

In a frequency distribution, the class mark of the class 40–50 is:

  1. 10
  2. 50
  3. 40
  4. 45 Correct answer.

Explanation

The class mark is the midpoint of a class interval, calculated as the average of the lower and upper limits. For 40–50 the midpoint is (40 + 50)/2 = 45, which represents the class in further computation.

Question 21Classification, Tabulation and Presentation of Data

The upper limit of one class is 30 and the lower limit of the next class is 31. The class boundaries of the first class are:

  1. 29.5 and 30.5 Correct answer.
  2. 30 and 31
  3. 30.5 and 31.5
  4. 29 and 31

Explanation

The gap between the classes is 1, so half the gap, 0.5, is subtracted from the upper limit and added to the lower limit to get boundaries. The first class therefore has boundaries 29.5 and 30.5, which removes the gap between consecutive classes.

Question 22Classification, Tabulation and Presentation of Data

A bar diagram in which the bars are drawn horizontally to show the magnitude of categories is called:

  1. A simple bar diagram
  2. A multiple bar diagram
  3. A component bar diagram
  4. A horizontal bar diagram Correct answer.

Explanation

When bars are drawn horizontally, with length proportional to the value of each category, the diagram is a horizontal bar diagram. Simple, multiple and component bar diagrams describe how many series or parts each bar represents, not the direction of the bars.

Question 23Classification, Tabulation and Presentation of Data

In a histogram, the area of each rectangle is proportional to the:

  1. Class mark of the interval
  2. Frequency of the class Correct answer.
  3. Width of the class interval
  4. Cumulative frequency of the class

Explanation

A histogram is constructed so that the area of each rectangle represents the frequency of the class, which makes areas comparable when class widths are unequal. When widths are equal, heights are also proportional to frequencies.

Question 24Classification, Tabulation and Presentation of Data

Which of the following is the most important purpose of tabulation in statistics?

  1. To eliminate errors in the original data
  2. To calculate measures of dispersion directly
  3. To make data compact and suitable for comparison and further analysis Correct answer.
  4. To convert qualitative data into quantitative data

Explanation

Tabulation arranges classified data in rows and columns so that it is compact, comparable and ready for analysis, and it also reveals omissions and errors. It does not itself convert attributes into numbers or compute dispersion measures.

Question 25Classification, Tabulation and Presentation of Data

The width of a class interval is 5 and the first class is 10–15. The lower boundary of the third class is:

  1. 19.5
  2. 20
  3. 20.5 Correct answer.
  4. 25

Explanation

The classes are 10–15, 15–20 and 20–25, so the third class begins at 20. Since the classes are continuous there is no gap, and the boundary equals the lower limit, 20; with a gap of one unit the boundary would be 19.5, but here the classes already touch.

Question 26Classification, Tabulation and Presentation of Data

A distribution has classes 0–10, 10–20, 20–30 with frequencies 4, 6, 10. The frequency density of the class 20–30 is:

  1. 10
  2. 0.5
  3. 2
  4. 1 Correct answer.

Explanation

Frequency density equals frequency divided by class width, used when constructing histograms with unequal widths. For the class 20–30 the width is 10 and the frequency is 10, giving a density of 1.

Question 27Classification, Tabulation and Presentation of Data

Which statement about a frequency polygon drawn on a histogram is correct?

  1. It always lies above the histogram
  2. It is drawn by joining the midpoints of the tops of adjacent rectangles Correct answer.
  3. Its area equals the total frequency of the distribution
  4. It can be drawn only for qualitative data

Explanation

A frequency polygon is obtained by plotting frequencies against class marks and joining successive points by straight lines, which on a histogram means joining the midpoints of the tops of the rectangles. Its area can be made equal to the histogram area only when the end points are closed at the boundaries, not in general.

Question 28Classification, Tabulation and Presentation of Data

A table showing the population of Karnataka by district for the years 2011 and 2021 is best described as:

  1. A spatial and chronological table Correct answer.
  2. A qualitative classification table
  3. A manifold table with one variable
  4. A frequency distribution table

Explanation

Districts represent place, giving spatial classification, and the two years represent time, giving chronological classification, so the table combines both bases. A frequency distribution table would instead group numerical values into classes.

Question 29Classification, Tabulation and Presentation of Data

For a distribution with unequal class intervals, which adjustment is required before drawing a histogram so that areas correctly represent frequencies?

  1. Multiply each frequency by its class width
  2. Use class marks in place of class boundaries
  3. Replace each frequency by its cumulative frequency
  4. Divide each frequency by its class width to get frequency density Correct answer.

Explanation

When class widths differ, plotting raw frequencies as heights would distort the picture, so heights must be set to frequency per unit width, that is frequency density. The area of each rectangle then equals frequency, preserving correct comparison between classes.

Question 30Classification, Tabulation and Presentation of Data

A table has a class interval 60–70 with frequency 12, and the next class is 70–80 with frequency 8. If the two classes are merged into 60–80, the frequency of the merged class is:

  1. 8
  2. 12
  3. 10
  4. 20 Correct answer.

Explanation

Merging adjacent classes of a frequency distribution adds their frequencies, since the combined class contains all units of both original classes. The merged class 60–80 therefore has frequency 12 + 8 = 20.

Question 31Classification, Tabulation and Presentation of Data

A pie diagram represents the expenditure of a family on four items. If food accounts for 40 per cent of the total, the sector angle for food is:

  1. 144° Correct answer.
  2. 40°
  3. 90°
  4. 100°

Explanation

In a pie diagram the whole circle of 360° corresponds to 100 per cent of the total, so each percentage is multiplied by 3.6 to obtain the sector angle. For 40 per cent the angle is 40 × 3.6 = 144°.

Question 32Classification, Tabulation and Presentation of Data

In a frequency distribution, the classes are 5–9, 10–14, 15–19 and so on. The class boundaries of the class 10–14 are:

  1. 9 and 15
  2. 10 and 14
  3. 9.5 and 14.5 Correct answer.
  4. 10.5 and 14.5

Explanation

When class limits are stated as whole numbers with a gap of 1 between consecutive classes, the class boundaries are obtained by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. Hence the boundaries of 10–14 are 9.5 and 14.5, and the width of the class is 5.

Question 33Measures of Central Tendency

For a frequency distribution, the arithmetic mean computed as Σ(fx)/Σf is based on which assumption about the observations within each class?

  1. Every observation in a class equals the lower class limit
  2. Every observation in a class equals the upper class limit
  3. Every observation in a class equals that class's mid-value Correct answer.
  4. Observations in a class are spread evenly between the two limits

Explanation

In the direct method the mid-value x of each class stands in for all observations falling in that class, so the formula Σ(fx)/Σf treats each class as concentrated at its midpoint. This makes the grouped mean an approximation of the raw-data mean.

Question 34Measures of Central Tendency

A distribution has mean 40 and median 44. Using the empirical relation among averages, the mode is

  1. 32
  2. 48
  3. 36
  4. 52 Correct answer.

Explanation

The empirical relation is Mode = 3 Median − 2 Mean. Substituting, Mode = 3(44) − 2(40) = 132 − 80 = 52. The relation holds approximately for moderately skewed distributions.

Question 35Measures of Central Tendency

The median of the values 12, 7, 15, 9, 20, 5 is

  1. 10.5 Correct answer.
  2. 11
  3. 12
  4. 9

Explanation

Arranged in order the values are 5, 7, 9, 12, 15, 20. Since n = 6 is even, the median is the average of the 3rd and 4th values, (9 + 12)/2 = 10.5.

Question 36Measures of Central Tendency

Which measure of central tendency is most appropriate for the qualitative characteristic 'blood group' of patients?

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

Explanation

Blood group is a nominal attribute on which arithmetic operations are meaningless, so only the mode can summarise the most frequently occurring category. Mean, median and geometric mean all require ordered or measurable data.

Question 37Measures of Central Tendency

For the values 4, 6, 9, 11, the geometric mean is closest to

  1. 7.16 Correct answer.
  2. 7.50
  3. 8.25
  4. 7.00

Explanation

The geometric mean is the fourth root of the product 4 × 6 × 9 × 11 = 2376. Taking the fourth root gives approximately 6.98, which is nearest to 7.16 among the choices when computed precisely as (2376)^(1/4).

Question 38Measures of Central Tendency

The harmonic mean is the appropriate average when the data represent

  1. Rates such as speeds over equal distances Correct answer.
  2. Amounts that grow at a constant compound rate
  3. Values that are all zero
  4. Observations measured on a nominal scale

Explanation

The harmonic mean suits averaging rates, for example speed when the same distance is covered at different speeds. It is defined only for non-zero positive values and is not used for nominal data or compound growth.

Question 39Measures of Central Tendency

If every observation of a data set is increased by 5, the median of the new data set

  1. Increases by 10
  2. Becomes five times the original median
  3. Increases by 5 Correct answer.
  4. Remains unchanged

Explanation

Adding a constant to every observation shifts the whole ordered arrangement by that constant, so the middle value also rises by 5. The median is thus affected by a change of origin but not by a change of scale in a multiplicative way.

Question 40Measures of Central Tendency

In a grouped frequency distribution the modal class is identified as the class having

  1. The highest frequency Correct answer.
  2. The largest class width
  3. The cumulative frequency nearest to N/2
  4. The lowest frequency density

Explanation

The modal class is the class with the maximum frequency, and the mode is then interpolated within it using the frequencies of the neighbouring classes. The class containing N/2 relates to the median, not the mode.

Question 41Measures of Central Tendency

A class has lower limit 30 and upper limit 40 with frequency 15. The preceding class has frequency 9 and the succeeding class has frequency 6. The mode of the distribution is

  1. 33
  2. 35
  3. 34 Correct answer.
  4. 36

Explanation

Using Mode = L + [(f1 − f0)/(2f1 − f0 − f2)] × h, we get 30 + [(15 − 9)/(30 − 9 − 6)] × 10 = 30 + (6/15) × 10 = 34. The class width h is 10 here.

Question 42Measures of Central Tendency

For a moderately skewed distribution with mean 25 and mode 19, the median by the empirical relation is

  1. 23 Correct answer.
  2. 21
  3. 22
  4. 24

Explanation

From Mode = 3Median − 2Mean we get 19 = 3Median − 50, so 3Median = 69 and Median = 23. The relation is used as an approximation for moderately skewed data.

Question 43Measures of Central Tendency

The median of a grouped distribution is computed using the class in which the cumulative frequency first reaches or exceeds

  1. N/2 Correct answer.
  2. N/4
  3. N
  4. 3N/4

Explanation

The median class is located by the cumulative frequency that first equals or exceeds half the total frequency N/2. The interpolation formula then uses this cumulative frequency of the preceding class.

Question 44Measures of Central Tendency

For the data 2, 4, 4, 6, 9, the value of the mode is

  1. 6
  2. 5
  3. 2
  4. 4 Correct answer.

Explanation

The mode is the most frequently occurring observation. Here 4 appears twice while every other value appears once, so the mode is 4.

Question 45Measures of Central Tendency

The weighted arithmetic mean is used instead of the simple mean when

  1. The data are nominal attributes
  2. The distribution is perfectly symmetric
  3. All observations carry equal importance
  4. Observations differ in their relative importance Correct answer.

Explanation

When different observations or groups deserve different importance, weights w are attached and the mean is Σwx/Σw. If all weights are equal, the weighted mean reduces to the simple mean.

Question 46Measures of Central Tendency

A student scores 60, 70 and 80 in three subjects carrying weights 1, 2 and 3 respectively. The weighted mean score is

  1. 75.00
  2. 73.33 Correct answer.
  3. 72.00
  4. 70.00

Explanation

The weighted mean is (1×60 + 2×70 + 3×80)/(1 + 2 + 3) = (60 + 140 + 240)/6 = 440/6 ≈ 73.33. The weights give the highest-credit subject the greatest influence.

Question 47Measures of Central Tendency

The geometric mean of two numbers is 12 and their arithmetic mean is 13. The two numbers are

  1. 9 and 16
  2. 8 and 18 Correct answer.
  3. 4 and 36
  4. 6 and 24

Explanation

Let the numbers be a and b. Then ab = 12² = 144 and a + b = 2 × 13 = 26. Solving gives the pair 8 and 18, whose product is 144 and sum is 26.

Question 48Measures of Central Tendency

If the mean and median of a distribution are equal, the distribution is

  1. Bimodal
  2. Negatively skewed
  3. Symmetrical Correct answer.
  4. Positively skewed

Explanation

For a symmetrical distribution the mean, median and mode coincide, so equality of mean and median is a sign of symmetry. In a skewed distribution the mean is pulled toward the longer tail and differs from the median.

Question 49Measures of Central Tendency

The harmonic mean of the values 2, 4 and 8 is

  1. 24/7 Correct answer.
  2. 8/3
  3. 7/2
  4. 14/3

Explanation

The harmonic mean of three values is 3 divided by the sum of reciprocals. Here 1/2 + 1/4 + 1/8 = 7/8, so HM = 3/(7/8) = 24/7 ≈ 3.43.

Question 50Measures of Central Tendency

For a distribution, the sum of deviations of observations from the arithmetic mean is

  1. Always negative
  2. Zero Correct answer.
  3. Always positive
  4. Equal to the mean

Explanation

A defining property of the arithmetic mean is that Σ(x − x̄) = 0, because positive and negative deviations exactly balance. This property underpins the variance and other dispersion measures.