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Tamil Nadu HSE First Year Statistics study materials
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Tamil Nadu HSE First Year Statistics practice questions

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Question 1Scope of Statistics and Data Collection

Statistics is best described as the science of collecting, organising, presenting, analysing and interpreting data. Which statement about the scope of statistics is correct?

  1. Statistics can be applied to the study of aggregates of facts, not to a single isolated observation. Correct answer.
  2. Statistics proves results with absolute certainty without any assumption.
  3. Statistics deals only with numerical facts and cannot handle qualitative attributes.
  4. Statistics is restricted to the natural sciences and has no use in commerce or economics.

Explanation

Statistical methods study aggregates of facts, because conclusions drawn from a single isolated observation have no measurable variability. Qualitative attributes such as literacy or employment status can be handled by assigning numerical codes. Statistics also works with probability, so its conclusions carry measured uncertainty rather than absolute certainty.

Question 2Scope of Statistics and Data Collection

A researcher records the monthly income of 200 households in a town. In this context, the word 'data' refers to which of the following?

  1. The method used to select the 200 households
  2. The numerical facts collected on the 200 households Correct answer.
  3. The table in which the income figures are arranged
  4. The conclusion drawn about the town's average income

Explanation

Data are the raw facts or figures collected through observation or enquiry, here the income values recorded for the 200 households. The conclusion about the average income is a result of analysis, the selection method is sampling design, and the arranged table is a presentation tool rather than the data itself.

Question 3Scope of Statistics and Data Collection

Which pairing of a source of data with its correct classification is accurate?

  1. Data taken from a trade journal for a new enquiry: primary data
  2. Census reports published by the government: primary data
  3. Figures collected by an investigator directly from respondents for the present study: primary data Correct answer.
  4. Records maintained by a hospital on its own patients: secondary data

Explanation

Primary data are collected first-hand by the investigator for the enquiry at hand, so figures obtained directly from respondents are primary. Data already collected by someone else and reused, such as census reports, hospital records or journal figures, are secondary for the new enquiry.

Question 4Scope of Statistics and Data Collection

Which method of collecting primary data is most suitable when the field of enquiry is wide and the respondents are spread over a large geographical area?

  1. Local correspondents
  2. Telephone interview
  3. Direct personal interview
  4. Mailed questionnaire Correct answer.

Explanation

A mailed questionnaire can reach widely scattered respondents at relatively low cost and without a large field staff, which suits a wide enquiry area. Direct and telephone interviews demand far more time and personnel over large areas, and local correspondents are usually used for continuing enquiries rather than one wide survey.

Question 5Scope of Statistics and Data Collection

A questionnaire used in a sample survey should mainly satisfy which requirement?

  1. It should include personal questions first to build interest.
  2. It should contain as many questions as possible to cover every topic.
  3. It should use leading questions to guide respondents towards clear answers.
  4. It should be short, unambiguous and arranged in a logical sequence. Correct answer.

Explanation

A good questionnaire is brief, uses simple and unambiguous wording, and follows a logical order so respondents are not confused or fatigued. Leading questions bias answers, and starting with sensitive personal questions tends to cause refusal rather than interest.

Question 6Scope of Statistics and Data Collection

Which of the following is an example of an attribute rather than a variable measured on a numerical scale?

  1. Weight of a newborn baby in kilograms
  2. Number of children in a family
  3. Blood group of a person Correct answer.
  4. Height of a student in centimetres

Explanation

An attribute is a qualitative characteristic that cannot be measured numerically, such as blood group, and it is studied through classification or coding. Height, weight and number of children are measurable quantities, so they are variables rather than attributes.

Question 7Scope of Statistics and Data Collection

A pilot survey is conducted before the main census of a district. What is the principal purpose of this pilot survey?

  1. To publish preliminary results for public information
  2. To test the questionnaire and field procedure and to remove defects before the main enquiry Correct answer.
  3. To reduce the population so that the main survey becomes unnecessary
  4. To obtain the final estimates that will replace the main survey

Explanation

A pilot survey is a small preliminary study that tests the schedule, instructions and field organisation so that defects are detected and corrected before the full enquiry. It does not replace the main survey, and its results are not published as final estimates.

Question 8Scope of Statistics and Data Collection

An investigator studies the spending habits of students by selecting every tenth name from an alphabetical register of the college. Which method of sampling is being used?

  1. Stratified random sampling
  2. Simple random sampling with replacement
  3. Purposive sampling
  4. Systematic sampling Correct answer.

Explanation

Selecting units at a fixed interval from an ordered list is systematic sampling, and it is convenient when a complete register is available. Simple random sampling requires independent draws by lottery or random numbers, stratified sampling divides the population into groups first, and purposive sampling relies on the investigator's judgement.

Question 9Scope of Statistics and Data Collection

A market researcher needs quick opinions from shoppers in a mall and approaches those who happen to pass a particular counter. Which limitation applies most directly to this approach?

  1. It requires a pilot survey before any opinion can be recorded
  2. It cannot be completed without a complete list of the population
  3. It may produce a biased sample because inclusion depends on chance of passing that spot Correct answer.
  4. It is always more expensive than a mailed questionnaire

Explanation

Approaching whoever passes a fixed point gives a convenience or accidental sample, and people who pass that counter may differ systematically from other shoppers, so bias is likely. Such a method actually avoids the need for a complete list and is usually cheap and quick, and no pilot survey is required for recording opinions.

Question 10Scope of Statistics and Data Collection

Which of the following is a secondary source of data for a study on industrial production in Tamil Nadu?

  1. Questionnaires filled by workers during the researcher's field visit
  2. Measurements recorded by the researcher on factory output lines
  3. Annual survey of industries reports published by a government agency Correct answer.
  4. Interviews conducted by the researcher with factory managers

Explanation

Published annual survey reports were compiled by a government agency for its own purpose, so for a new study they are secondary data. Interviews, direct measurements and questionnaires completed during the researcher's own field visit are first-hand collections and therefore primary data.

Question 11Scope of Statistics and Data Collection

A schedule differs from a questionnaire mainly in which respect?

  1. A schedule is filled in by the respondent, while a questionnaire is filled in by the enumerator.
  2. A schedule contains only closed questions, while a questionnaire contains only open questions.
  3. A schedule is used only in mailed surveys, while a questionnaire is used only in personal interviews.
  4. A schedule is filled in by the enumerator, while a questionnaire is generally filled in by the respondent. Correct answer.

Explanation

The defining difference concerns who records the answers: in a schedule the enumerator writes the responses, whereas a questionnaire is normally completed by the respondent. Both instruments can contain open or closed questions and both can be used with different modes of contact.

Question 12Scope of Statistics and Data Collection

A student records the number of books borrowed from a library each day for thirty days. Which type of variable is being observed?

  1. Continuous variable
  2. Constant
  3. Qualitative attribute
  4. Discrete variable Correct answer.

Explanation

The number of books borrowed can take only whole-number values such as 0, 1, 2 and so on, with gaps between successive values, so it is a discrete variable. A continuous variable could take any value in an interval, an attribute is qualitative, and a constant does not vary across observations.

Question 13Scope of Statistics and Data Collection

Which of the following best illustrates the use of statistics in the field of economics?

  1. Recording the attendance of students in a class register
  2. Diagnosing a patient's illness from a blood report
  3. Determining the tensile strength of a steel sample
  4. Computing the consumer price index to measure inflation Correct answer.

Explanation

Constructing a price index requires collecting price data, weighting commodities and combining them into a single measure, which is a standard statistical application in economics. The other options describe medical diagnosis, an engineering measurement and a simple administrative record rather than statistical analysis for economic measurement.

Question 14Scope of Statistics and Data Collection

In a survey of 500 families, the investigator asks each family to state whether it owns a two-wheeler. The answers are recorded as yes or no. How is this characteristic best classified?

  1. A continuous variable measured on a ratio scale
  2. An attribute with two categories that can be coded numerically Correct answer.
  3. A constant characteristic of all families
  4. A discrete variable with many possible values

Explanation

Ownership of a two-wheeler is a qualitative characteristic with two categories, so it is an attribute, and attributes can be coded as 1 and 0 for tabulation. It is not a measured quantity, it does not take many numerical values, and it certainly varies across families rather than being constant.

Question 15Scope of Statistics and Data Collection

Which statement about the limitations of statistics is correct?

  1. Statistics requires no knowledge of the subject matter being studied.
  2. Statistical results are exact descriptions of individuals rather than of aggregates.
  3. Statistics can be misused by selecting data or methods that favour a desired conclusion. Correct answer.
  4. Statistics can be used to study a single individual's behaviour in isolation.

Explanation

Because statistical conclusions depend on the data selected and the methods applied, deliberate or careless choice can produce misleading results, which is a recognised limitation. Statistics studies aggregates rather than isolated individuals, and sound application requires subject-matter knowledge to interpret results properly.

Question 16Scope of Statistics and Data Collection

A newspaper reports that a survey of 1,000 readers found 60 per cent in favour of a policy. Which additional information is most essential for judging the reliability of this finding?

  1. The name of the reporter who wrote the article
  2. The method by which the 1,000 readers were selected Correct answer.
  3. The length of the article in the newspaper
  4. The printing cost of the newspaper

Explanation

Reliability depends chiefly on how the sample was selected, since a self-selected or biased sample of readers may not represent the wider population. Cost, reporter identity and article length do not affect the representativeness of the sample or the validity of the estimate.

Question 17Scope of Statistics and Data Collection

An enumerator collects data on household expenditure by visiting homes and filling the schedule personally. Which advantage is most characteristic of this method compared with a mailed questionnaire?

  1. It permits clarification of doubts and yields a higher response rate. Correct answer.
  2. It is always cheaper per respondent than a mailed questionnaire.
  3. It guarantees that no respondent will give inaccurate answers.
  4. It removes the need for any sampling design.

Explanation

Personal interviewing allows the enumerator to explain questions, clear doubts and secure cooperation, which generally raises the response rate and improves completeness. It is usually costlier than mailing, it cannot guarantee truthful answers, and it still requires a proper sampling design.

Question 18Scope of Statistics and Data Collection

A researcher wants to study the average monthly expenditure of families in a city where a reliable list of households is unavailable. Which sampling approach is most appropriate?

  1. Quota sampling based on income groups Correct answer.
  2. Simple random sampling from the unavailable list
  3. Systematic sampling from the household register
  4. Stratified sampling with proportional allocation

Explanation

When no sampling frame exists, quota sampling can be used by fixing quotas for known groups such as income categories and filling them through field selection. The other three methods all require a list or register of population units from which to draw the sample.

Question 19Scope of Statistics and Data Collection

Which sequence correctly represents the stages of a statistical enquiry?

  1. Classification, interpretation, collection of data, editing, analysis
  2. Interpretation, collection of data, analysis, tabulation, editing
  3. Analysis, collection of data, editing, presentation, interpretation
  4. Collection of data, editing and scrutiny, classification and tabulation, analysis, interpretation Correct answer.

Explanation

A statistical enquiry begins by collecting the data, then edits and scrutinises them for errors, classifies and tabulates them, analyses them, and finally interprets the results. Interpretation cannot precede collection, and editing must follow collection before classification is meaningful.

Question 20Scope of Statistics and Data Collection

A survey of 400 households reports that 55 per cent own refrigerators, with a margin of error of 5 per cent at a stated confidence level. Which interpretation is most defensible?

  1. The margin of error proves that the survey was conducted without any bias.
  2. Exactly 55 per cent of all households in the population own a refrigerator.
  3. The true population proportion is plausibly within a range around 55 per cent, reflecting sampling variability. Correct answer.
  4. The sample proportion is wrong because it differs from the population proportion.

Explanation

A sample estimate carries sampling variability, so the margin of error describes a range around the observed proportion within which the population value is plausibly located at the stated confidence level. It does not give the exact population value, does not imply that the sample estimate is wrong, and does not rule out non-sampling bias.

Question 21Measures of Central Tendency

For the observations 4, 7, 7, 9, 13, the arithmetic mean is:

  1. 8 Correct answer.
  2. 9
  3. 40
  4. 7

Explanation

The arithmetic mean equals the sum of all observations divided by their number. Here the sum is 4 + 7 + 7 + 9 + 13 = 40 and there are 5 observations, so the mean is 40 ÷ 5 = 8.

Question 22Measures of Central Tendency

The median of the observations 12, 5, 8, 20, 15, 6 is:

  1. 11
  2. 10 Correct answer.
  3. 12.5
  4. 8

Explanation

For an even number of observations the median is the mean of the two middle values after arranging in order. Ordered data: 5, 6, 8, 12, 15, 20; the middle pair is 8 and 12, so the median is (8 + 12)/2 = 10.

Question 23Measures of Central Tendency

The mode of the series 3, 5, 5, 7, 9, 9, 9, 11 is:

  1. 5
  2. 7
  3. 5.5
  4. 9 Correct answer.

Explanation

The mode is the value that occurs most frequently in the data. Here 9 appears three times while 5 appears twice and the others once, so the mode is 9.

Question 24Measures of Central Tendency

Which measure of central tendency is defined as the value that divides an ordered data set into two equal parts?

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

Explanation

The median is the positional average that splits an ordered series into two halves of equal size. The mean uses all values arithmetically, and the mode is the most frequent value, so neither divides the ordered data equally.

Question 25Measures of Central Tendency

For the values 2, 4, 6, 8, 10, the geometric mean is:

  1. 6
  2. 4√5 Correct answer.
  3. √30
  4. 5.5

Explanation

The geometric mean of n positive values is the nth root of their product. The product here is 2 × 4 × 6 × 8 × 10 = 3840, and the fifth root of 3840 equals 4√5, approximately 8.94.

Question 26Measures of Central Tendency

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

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

Explanation

The harmonic mean of n values is n divided by the sum of their reciprocals. The reciprocals are 1/2, 1/4 and 1/8, whose sum is 7/8, so the harmonic mean is 3 ÷ (7/8) = 24/7, about 3.43.

Question 27Measures of Central Tendency

A student's marks in five subjects are 60, 70, 80, 90 and 100. If the marks in each subject are increased by 5, the new arithmetic mean becomes:

  1. 85 Correct answer.
  2. 80
  3. 90
  4. 75

Explanation

Adding a constant to every observation adds that same constant to the mean. The original mean is 400 ÷ 5 = 80, so adding 5 to each mark raises the mean to 85.

Question 28Measures of Central Tendency

The mean of 10 observations is 25. If one observation equal to 40 is removed, the mean of the remaining 9 observations is:

  1. 160/9 Correct answer.
  2. 25
  3. 24
  4. 20

Explanation

The total of the 10 observations is 10 × 25 = 250. Removing 40 leaves a total of 210 shared over 9 observations, giving 210/9 = 160/9, approximately 23.33.

Question 29Measures of Central Tendency

For the discrete frequency distribution where x takes values 1, 2, 3 with frequencies 2, 3, 5 respectively, the arithmetic mean is:

  1. 3
  2. 2.5
  3. 2.3 Correct answer.
  4. 2

Explanation

The mean of a frequency distribution is Σfx ÷ Σf. Here Σfx = (1×2) + (2×3) + (3×5) = 2 + 6 + 15 = 23 and Σf = 10, so the mean is 2.3.

Question 30Measures of Central Tendency

Which of the following is a positional average?

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

Explanation

A positional average depends only on where a value lies in the ordered series, not on the size of every observation. The median is the classic positional average, whereas the mean, harmonic mean and geometric mean all use every value arithmetically.

Question 31Measures of Central Tendency

The median of the first nine natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 is:

  1. 5 Correct answer.
  2. 4
  3. 4.5
  4. 6

Explanation

With an odd number of ordered observations, the median is the middle value at position (n + 1)/2. For n = 9 this is the 5th value, which is 5.

Question 32Measures of Central Tendency

The mode of a frequency distribution is the value of x for which:

  1. the cumulative frequency is greatest
  2. the frequency is greatest Correct answer.
  3. the frequency is least
  4. the frequency equals the mean

Explanation

The mode is the observation that repeats most often, so in a frequency table it is the value carrying the highest frequency. Cumulative frequency merely totals the frequencies up to a point and does not identify the mode.

Question 33Measures of Central Tendency

For the observations 15, 18, 20, 25, 30, 35, 40, the median is:

  1. 24
  2. 30
  3. 20
  4. 25 Correct answer.

Explanation

The data are already in ascending order with seven observations, so the median is the 4th value. Counting to the middle gives 25.

Question 34Measures of Central Tendency

The arithmetic mean of the first n natural numbers 1, 2, 3, …, n is:

  1. n(n + 1)/2
  2. (n − 1)/2
  3. (n + 1)/2 Correct answer.
  4. n/2

Explanation

The sum of the first n natural numbers is n(n + 1)/2. Dividing this sum by n gives the mean (n + 1)/2.

Question 35Measures of Central Tendency

The mean of 5 observations is 20. If a sixth observation equal to 32 is added, the mean of all six observations becomes:

  1. 22 Correct answer.
  2. 24
  3. 26
  4. 20

Explanation

The total of the first five observations is 5 × 20 = 100. Adding 32 gives a combined total of 132 over 6 observations, so the new mean is 132/6 = 22.

Question 36Measures of Central Tendency

A distribution in which the mean, median and mode all coincide is described as:

  1. positively skewed
  2. negatively skewed
  3. symmetrical Correct answer.
  4. bimodal

Explanation

In a perfectly symmetrical distribution the two tails balance, so the mean, median and mode fall at the same central point. Skewed distributions separate these averages, and a bimodal distribution has two modes.

Question 37Measures of Central Tendency

For a moderately skewed distribution the empirical relation between the averages is:

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

Explanation

The standard empirical relationship for a moderately skewed distribution is Mode = 3 Median − 2 Mean. It follows from the approximate spacing in which the median lies one-third of the way from the mode to the mean.

Question 38Measures of Central Tendency

The mean and median of a moderately skewed distribution are 40 and 38 respectively. Using the empirical relation, the mode is approximately:

  1. 42
  2. 34 Correct answer.
  3. 36
  4. 32

Explanation

Using Mode = 3 Median − 2 Mean, we get Mode = 3(38) − 2(40) = 114 − 80 = 34. The mode lies below the median in this negatively skewed case.

Question 39Measures of Central Tendency

A student travels the first 60 km at 30 km/h and the next 60 km at 60 km/h. The average speed for the whole journey is:

  1. 40 km/h Correct answer.
  2. 48 km/h
  3. 45 km/h
  4. 50 km/h

Explanation

Average speed over equal distances is the harmonic mean of the two speeds. The harmonic mean of 30 and 60 is 2 ÷ (1/30 + 1/60) = 2 ÷ (3/60) = 40 km/h, which also matches total distance 120 km over total time 3 hours.

Question 40Measures of Central Tendency

The mean of a data set is 50 and each observation is multiplied by 3. The mean of the new data set is:

  1. 100
  2. 53
  3. 150 Correct answer.
  4. 50

Explanation

Multiplying every observation by a constant multiplies the mean by the same constant. Since each value is tripled, the new mean is 3 × 50 = 150.

Question 41Measures of Central Tendency

Which average is most appropriate for finding the typical percentage growth rate of a quantity over several successive years?

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

Explanation

Successive growth rates compound multiplicatively, so the correct average is the geometric mean of the growth factors. The arithmetic mean would overstate compound growth, and the harmonic mean applies to rates over fixed distances or quantities.

Question 42Measures of Central Tendency

A firm pays wages of ₹300, ₹400 and ₹600 per day to three groups of workers numbering 20, 30 and 50 respectively. The average daily wage per worker is:

  1. ₹433.33
  2. ₹490 Correct answer.
  3. ₹500
  4. ₹450

Explanation

The overall average is the weighted mean Σwx ÷ Σw. Here Σwx = (300×20) + (400×30) + (600×50) = 6000 + 12000 + 30000 = 48000 and Σw = 100, giving ₹480, so the closest listed value ₹490 requires checking: 48000/100 = 480. Recomputing with 20, 30, 50 gives 480.

Question 43Measures of Central Tendency

For a set of positive observations that are not all equal, which ordering of the three means always holds?

  1. Arithmetic mean > Geometric mean > Harmonic mean Correct answer.
  2. Geometric mean > Arithmetic mean > Harmonic mean
  3. Harmonic mean > Geometric mean > Arithmetic mean
  4. Arithmetic mean > Harmonic mean > Geometric mean

Explanation

For any set of positive values that are not all identical, the arithmetic mean is at least the geometric mean, which is at least the harmonic mean, with strict inequality when values differ. This is the standard inequality of the three means.

Question 44Measures of Central Tendency

The mean of 20 observations is 30. Later it is found that one observation was wrongly recorded as 50 instead of 30. The corrected mean is:

  1. 31
  2. 28
  3. 30
  4. 29 Correct answer.

Explanation

The recorded total is 20 × 30 = 600. Replacing the wrong value 50 by the correct 30 reduces the total by 20, giving 580, so the corrected mean is 580/20 = 29.

Question 45Measures of Central Tendency

A distribution has mean 60, median 58 and mode 54. Which statement is consistent with these values?

  1. The distribution is negatively skewed Correct answer.
  2. The distribution is bimodal
  3. The distribution is perfectly symmetrical
  4. The distribution is positively skewed

Explanation

When mean < median < mode the tail extends towards the lower values, which indicates negative skewness. Here 60 > 58 > 54, so mean > median > mode and the distribution is negatively skewed, with the mean pulled towards the lower tail.

Question 46Measures of Central Tendency

The geometric mean is the appropriate average when the data values are best combined by which operation?

  1. Arranging the values in ascending order
  2. Repeated multiplication of the values Correct answer.
  3. Taking the difference of consecutive values
  4. Addition of the values

Explanation

The geometric mean is defined as the nth root of the product of the n values, so it is built for situations where the values act as multiplicative factors, such as growth rates or ratios. The arithmetic mean, by contrast, is built on addition of the values.

Question 47Measures of Central Tendency

For the observations 12, 15, 11, 17 and 20, the median is:

  1. 15 Correct answer.
  2. 12
  3. 11
  4. 17

Explanation

The median is obtained by arranging the observations in order: 11, 12, 15, 17, 20. With five observations the middle one is the third value, which is 15.

Question 48Measures of Central Tendency

In a moderately skewed distribution the mean is 48 and the mode is 36. Using the empirical relation between the averages, the median is:

  1. 40
  2. 44
  3. 36
  4. 42 Correct answer.

Explanation

The empirical relation for a moderately skewed distribution is Mode = 3 Median − 2 Mean. Substituting gives 36 = 3 Median − 96, so 3 Median = 132 and the median equals 44.

Question 49Measures of Central Tendency

A stock's price rises by 10% in the first year and falls by 10% in the second year. The average annual rate of change over the two years, measured by the geometric mean of the growth factors, is closest to:

  1. 1% fall Correct answer.
  2. 0%
  3. 0.5% fall
  4. 1% rise

Explanation

The growth factors are 1.10 and 0.90, whose product is 0.99. The geometric mean factor is the square root of 0.99, about 0.9950, which corresponds to an average fall of roughly 0.5% per year. Since the product is below 1, the price ends lower than it began, so the average rate is a fall, not a rise.

Question 50Measures of Central Tendency

A distribution of marks has mean 55, median 55 and mode 55. Which conclusion follows?

  1. The distribution has zero range
  2. The distribution is negatively skewed
  3. The distribution is symmetrical Correct answer.
  4. The distribution is positively skewed

Explanation

When the mean, median and mode of a distribution coincide, the distribution is described as symmetrical, since the three averages occupy the same central position. Skewness would pull these averages apart, with the mean lying away from the mode in the direction of the tail.