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Table of Contents
Chapter 1: Introduction to Statistics
Statistics is the science of collecting, organising, presenting, analysing and interpreting numerical facts so that a sensible decision can be made from them. In everyday life we meet statistics in cricket averages, prices in a market, marks in a class and the number of patients in a hospital. The subject does not merely give us tables of numbers; it gives us a method for turning raw figures into clear information.
Important points
Statistics has two broad branches. Descriptive statistics describes and summarises data, for example by finding an average or drawing a chart. Inferential statistics uses a sample to draw a conclusion about a larger population. Data are the raw facts, while a population is the whole group under study and a sample is a part of that group chosen for study. A variable is a characteristic that changes from person to person, such as height or marks, while a constant stays the same. Data can be primary, collected first-hand by the investigator, or secondary, taken from an already published source. Statistics is useful in business, economics, medicine, education and government planning, but it must be used honestly: a wrong sample or a misleading graph can give a wrong conclusion.
Short questions with full detailed answers
Q1. What is statistics?
Answer: Statistics is the branch of knowledge that deals with the collection, classification, presentation, analysis and interpretation of numerical data. For example, if a teacher collects the marks of 40 students, arranges them in order, finds their average and draws a graph, all of that work is statistics. Its purpose is not to collect numbers for their own sake, but to help us understand a situation and make a reasoned decision.
Q2. Distinguish between a population and a sample.
Answer: A population is the complete set of items or persons about which information is required. For example, all first-year students in a college form a population. A sample is a smaller part of the population that is actually studied. For example, 50 students chosen from those 500 students form a sample. We study a sample because studying the whole population is often costly and time-consuming, and a well-chosen sample can still give a reliable picture of the population.
Q3. What is the difference between primary and secondary data?
Answer: Primary data are collected by the investigator himself for the first time for his own purpose, for example by a questionnaire or by direct interview. Secondary data are data already collected by someone else and published, for example in a government report or a newspaper. Primary data are usually more accurate for the purpose in hand but cost more time and money, while secondary data are cheaper and quicker but may not fit the purpose exactly.
Q4. Define a variable and give two examples.
Answer: A variable is a characteristic that can take different values for different individuals. For example, the height of students is a variable because one student may be 160 cm tall and another 172 cm tall. The marks obtained in a test are also a variable, because different students obtain different marks. A quantity that does not change, such as the number of days in a week, is a constant, not a variable.
Q5. Why is statistics important in daily life?
Answer: Statistics is important because decisions in daily life often depend on numbers. A shopkeeper studies past sales to decide how much stock to order, a doctor studies the recovery rate of patients to judge a treatment, and a government studies population figures to plan schools and hospitals. In each case statistics turns scattered figures into a summary, such as an average or a percentage, which makes comparison and planning possible. Without it, decisions would rest on guesswork alone.
Long questions with full detailed answers
Q1. Explain the scope and importance of statistics, with examples from different fields.
Answer: Introduction: Statistics touches almost every field of modern life because every field produces numbers that must be understood before a decision is made. Explanation: In business, a firm studies sales figures to decide production and advertising. In economics, price indices and national income figures guide policy. In medicine, recovery rates and the results of trials show whether a treatment works. In education, examination results show which topics students find difficult. In government, census figures guide the building of schools, roads and hospitals. Example: Suppose a shop records daily sales of 20, 25, 22, 30 and 28 packets over five days. The average is 25 packets a day, so the shopkeeper can order about 25 packets a day with confidence instead of guessing. Conclusion: Statistics is therefore not a collection of dry numbers; it is a practical tool for planning, comparison and decision-making in every field.
Q2. Describe the stages of a statistical investigation, from planning to conclusion.
Answer: Introduction: A statistical investigation is carried out in an orderly sequence of stages, and skipping a stage usually weakens the result. Explanation: First, the purpose is stated clearly, for example to find the average marks of first-year students. Second, the population is defined and a suitable sample is chosen. Third, data are collected by a questionnaire, interview or observation. Fourth, the data are classified and presented in tables or graphs. Fifth, the data are analysed, for example by finding averages or percentages. Sixth, the results are interpreted and a conclusion is drawn. Example: To study study habits, a student might plan the question, choose 40 classmates as a sample, collect their daily study hours, tabulate them, find that the average is 3 hours, and conclude that most students study between 2 and 4 hours a day. Conclusion: When these stages are followed carefully, the conclusion is trustworthy; when they are rushed, even correct arithmetic can support a wrong conclusion.
Multiple-choice questions with answer key
- Statistics is mainly concerned with:
(a) stories (b) numerical data (c) poetry (d) paintings - A sample is:
(a) the whole population (b) a part of the population (c) a constant (d) a graph - Data collected by the investigator himself are called:
(a) secondary data (b) primary data (c) grouped data (d) published data - The branch that only describes data is called:
(a) inferential statistics (b) descriptive statistics (c) probability (d) sampling - Which of the following is a variable?
(a) number of days in a week (b) height of students (c) number of months in a year (d) number of hours in a day
Answer key: 1-(b), 2-(b), 3-(b), 4-(b), 5-(b).
Chapter 2: Presentation of Data
Raw data are difficult to read and compare, so they must be presented in a clear form. Presentation means arranging data in tables and diagrams so that the main facts stand out at a glance. A good presentation saves time, makes comparison easy and helps the reader see a pattern, such as whether values are rising or falling.
Important points
Data are first classified, that is, grouped according to a common quality or size. A frequency distribution shows how many times each value occurs. Class intervals group values into ranges, such as 10-19, 20-29, and the mid-point of a class is called the class mark. Common diagrams are the bar chart, which uses bars of equal width, the histogram, which uses bars joined together for continuous data, the frequency polygon, which joins the mid-points of histogram bars, and the pie chart, which shows parts of a whole as sectors of a circle. A good table has a clear title, column headings, units and a source note. The choice of diagram matters: a bar chart suits separate categories, while a histogram suits grouped continuous data.
Short questions with full detailed answers
Q1. What is meant by presentation of data?
Answer: Presentation of data means arranging collected data in a systematic and attractive form, such as a table, chart or graph, so that it can be understood quickly. For example, the marks of 30 students written in a random list are hard to judge, but the same marks arranged in a frequency table, with the number of students in each marks range, at once show where most students fall. Presentation therefore turns raw figures into readable information.
Q2. What is a frequency distribution?
Answer: A frequency distribution is a table that shows each value, or each class of values, together with the number of times it occurs, called its frequency. For example, if in a class of 20 students, 5 students scored 60-69 marks and 8 scored 70-79 marks, those counts are the frequencies of those classes. A frequency distribution summarises a large set of data in a few rows and is the basis for most diagrams.
Q3. Distinguish between a bar chart and a histogram.
Answer: A bar chart represents separate categories, such as subjects or cities, by bars of equal width separated by gaps, and the height of each bar shows its value. A histogram represents continuous grouped data, such as heights or marks, by bars that touch each other, with no gaps, and the area of each bar is related to its frequency. In short, gaps and separate categories mark a bar chart, while touching bars and continuous classes mark a histogram.
Q4. What is a pie chart, and when is it used?
Answer: A pie chart is a circle divided into sectors, where each sector represents a part of the whole. The angle of a sector is found by the formula: angle = (value / total) x 360 degrees. For example, if out of 40 students, 10 prefer cricket, the cricket sector has angle (10/40) x 360 = 90 degrees. A pie chart is used when we want to compare parts with the whole, such as how a monthly budget is shared among food, rent and transport.
Q5. What qualities should a good table have?
Answer: A good table should have a clear and short title stating what it shows, well-labelled rows and columns, the unit of measurement, such as marks or rupees, and figures arranged neatly so that totals can be checked. For example, a table of monthly rainfall should state the place, the months, the unit in millimetres and the source of the figures. Without a title, headings and units, even correct figures can be misunderstood.
Long questions with full detailed answers
Q1. Explain, with examples, how raw data are converted into a frequency distribution.
Answer: Introduction: Raw data in their original order are hard to interpret, so they are grouped into a frequency distribution. Explanation: First the range is found by subtracting the smallest value from the largest. Then a suitable class interval is chosen, for example 10, and classes are formed such as 10-19, 20-29, 30-39. Each observation is then tallied against its class, and the tally count gives the frequency of that class. Example: For the illustrative marks 12, 18, 22, 25, 31, 35 and 38, the classes 10-19, 20-29 and 30-39 have frequencies 2, 2 and 3. The table at once shows that most values lie in the highest class. Conclusion: A frequency distribution therefore replaces a long list with a compact summary that can be used for averages, graphs and comparison.
Q2. Compare the bar chart, histogram, frequency polygon and pie chart, and state when each is best used.
Answer: Introduction: Different diagrams suit different kinds of data, and choosing the wrong one can mislead the reader. Explanation: A bar chart uses separate bars for separate categories, such as the number of students in each subject, and is easy to read. A histogram uses touching bars for continuous grouped data, such as heights, and shows the shape of the distribution. A frequency polygon joins the mid-points of the tops of histogram bars and is useful for comparing two distributions on one graph. A pie chart divides a circle into sectors and is best for showing how a whole is shared among parts, such as a budget. Example: To show sales by four separate shops, a bar chart is clearest; to show how a family spends its total income, a pie chart is clearest. Conclusion: The diagram should match the data: separate categories call for bars, continuous data call for a histogram, comparison calls for a polygon, and parts of a whole call for a pie chart.
Multiple-choice questions with answer key
- A frequency is:
(a) the largest value (b) the number of times a value occurs (c) the class mark (d) the total - In a histogram, the bars:
(a) are separated by gaps (b) touch each other (c) are all equal (d) are circles - The angle of a sector in a pie chart is found by:
(a) value x 360 (b) (value / total) x 360 (c) total / value (d) value + 360 - The mid-point of the class 20-29 is:
(a) 20 (b) 24.5 (c) 29 (d) 9 - A bar chart is most suitable for:
(a) continuous heights (b) separate categories (c) parts of a whole only (d) probabilities
Answer key: 1-(b), 2-(b), 3-(b), 4-(b), 5-(b).
Chapter 3: Measures of Central Tendency
A measure of central tendency is a single value that represents the centre, or typical value, of a set of data. Instead of listing all values, we can often summarise them by one number, such as the average marks of a class. The three main measures are the mean, the median and the mode, and each is useful in a different situation.
Important points
The arithmetic mean is found by adding all values and dividing by the number of values. For the values 4, 6, 8, 10 and 12, the sum is 40 and there are 5 values, so the mean is 8. The median is the middle value when data are arranged in order; for an even number of values, it is the average of the two middle values. The mode is the value that occurs most often. The mean uses all values but is affected by very large or very small values, while the median is not. For grouped data, the mean is found using class marks and frequencies. Choosing the right average matters: income is often described by the median, because a few very high incomes would raise the mean misleadingly.
Short questions with full detailed answers
Q1. Define the arithmetic mean and find it for 5, 10, 15 and 20.
Answer: The arithmetic mean is the sum of all observations divided by the number of observations. For 5, 10, 15 and 20, the sum is 5 + 10 + 15 + 20 = 50, and there are 4 observations, so the mean is 50 / 4 = 12.5. The mean gives the value each observation would have if the total were shared equally.
Q2. How is the median found for an odd and for an even number of values?
Answer: First arrange the values in ascending order. If the number of values is odd, the median is the middle value. For example, for 3, 5, 7, 9 and 11, the middle value is 7, so the median is 7. If the number is even, the median is the average of the two middle values. For example, for 3, 5, 7 and 9, the two middle values are 5 and 7, so the median is (5 + 7) / 2 = 6.
Q3. What is the mode? Give an example.
Answer: The mode is the value that occurs most frequently in a set of data. For example, in the values 2, 4, 4, 4, 7 and 9, the value 4 occurs three times, more than any other value, so the mode is 4. A set of data can have more than one mode, or no mode at all if every value occurs only once.
Q4. Why is the median sometimes better than the mean?
Answer: The median is sometimes better because it is not affected by extreme values. For example, for the incomes 20, 22, 24, 26 and 200, the mean is 58.4, which is higher than most of the incomes and gives a misleading picture, while the median is 24, which is typical of most of the group. When data contain a few very large or very small values, the median describes the typical value more honestly.
Q5. Find the mean of the grouped data: class marks 15, 25, 35 with frequencies 2, 3 and 5.
Answer: For grouped data, multiply each class mark by its frequency and add the results, then divide by the total frequency. Here, 15 x 2 = 30, 25 x 3 = 75 and 35 x 5 = 175. The sum is 30 + 75 + 175 = 280. The total frequency is 2 + 3 + 5 = 10. Therefore the mean is 280 / 10 = 28.
Long questions with full detailed answers
Q1. Find and compare the mean, median and mode of the illustrative values 10, 12, 14, 14, 18 and 22, and explain what each tells us.
Answer: Introduction: A single set of data can be summarised in three different ways, and comparing them shows different features of the data. Explanation: The mean is the sum divided by the count: 10 + 12 + 14 + 14 + 18 + 22 = 90, and 90 / 6 = 15. The median, for six ordered values, is the average of the third and fourth values: (14 + 14) / 2 = 14. The mode is the most frequent value, which is 14, occurring twice. Example: The mean of 15 is slightly above the median and mode of 14 because the larger values 18 and 22 pull the mean upward. Conclusion: Together the three measures show that the typical value is about 14 to 15, with the mean a little higher because of the larger values at the top end.
Q2. Explain the merits and demerits of the mean, median and mode, and state when each should be used.
Answer: Introduction: No single average is best for every situation, so a student must know the strengths and weaknesses of each. Explanation: The mean uses every value and is easy to work with in further calculations, but it is affected by extreme values and cannot be found for open-ended classes. The median is not affected by extreme values and can be found even for open-ended data, but it ignores most values and is harder to use in further algebra. The mode is simple, is not affected by extreme values and suits qualities such as the most popular size of shoe, but it may not exist, or there may be several modes. Example: For average income, the median is usually preferred; for average marks in a test, the mean is usually preferred; for the most common shoe size in a shop, the mode is preferred. Conclusion: The choice of average should follow the purpose: use the mean for a balanced summary, the median for skewed data, and the mode for the most frequent case.
Multiple-choice questions with answer key
- The mean of 4, 6, 8, 10 and 12 is:
(a) 6 (b) 8 (c) 10 (d) 12 - The median of 3, 5, 7, 9 and 11 is:
(a) 5 (b) 7 (c) 9 (d) 3 - The mode of 2, 4, 4, 4, 7 and 9 is:
(a) 2 (b) 7 (c) 4 (d) 9 - Which average is most affected by an extreme value?
(a) median (b) mode (c) mean (d) none - For grouped data, the mean uses:
(a) class marks and frequencies (b) only frequencies (c) only class limits (d) the mode
Answer key: 1-(b), 2-(b), 3-(c), 4-(c), 5-(a).
Chapter 4: Measures of Dispersion
An average alone does not describe data fully, because two sets can have the same mean but very different spreads. Dispersion measures how far values lie from the centre. A small dispersion means values are close together and the average is reliable; a large dispersion means values are widely spread and the average is less reliable.
Important points
The range is the simplest measure: largest value minus smallest value. The mean deviation is the average of the absolute deviations from the mean, ignoring signs. The variance is the average of the squared deviations from the mean, and the standard deviation is the square root of the variance. The standard deviation is the most widely used measure because it uses every value and is expressed in the same units as the data. The coefficient of variation, found by dividing the standard deviation by the mean and multiplying by 100, allows comparison of the spread of two sets measured in different units. Quartile deviation, based on the first and third quartiles, is less affected by extreme values.
Short questions with full detailed answers
Q1. What is dispersion, and why is it studied with an average?
Answer: Dispersion is the spread or scatter of values around a central value. It is studied with an average because the average alone can hide differences. For example, two classes may both have a mean of 60 marks, but in one class most students score between 55 and 65, while in the other scores range from 20 to 100. The second class is much more dispersed, so its mean is a less reliable summary. Dispersion therefore tells us how trustworthy the average is.
Q2. Find the range of 8, 12, 15, 20 and 27.
Answer: The range is the largest value minus the smallest value. Here the largest value is 27 and the smallest is 8, so the range is 27 – 8 = 19. The range is simple to find, but it uses only two values and is affected by a single extreme value.
Q3. Define variance and standard deviation, and state their relation.
Answer: The variance is the mean of the squared deviations of the values from their mean. The standard deviation is the positive square root of the variance. In symbols, if variance is denoted by sigma squared, then standard deviation is sigma. For example, if the variance is 16, the standard deviation is 4. The standard deviation is preferred in practice because it is in the same units as the original data, while variance is in squared units.
Q4. Find the standard deviation of 2, 4, 6, 8 and 10.
Answer: First find the mean: (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6. The deviations from the mean are -4, -2, 0, 2 and 4. Squaring them gives 16, 4, 0, 4 and 16, whose sum is 40. The variance is 40 / 5 = 8. The standard deviation is the square root of 8, which is about 2.83. This shows the values typically lie about 2.83 units from the mean.
Q5. What is the coefficient of variation, and when is it used?
Answer: The coefficient of variation is the standard deviation divided by the mean, multiplied by 100, giving a percentage. For example, if the mean is 50 and the standard deviation is 5, the coefficient of variation is (5 / 50) x 100 = 10 percent. It is used to compare the spread of two sets of data that have different means or different units, because a plain standard deviation cannot be compared fairly in such cases. The set with the larger coefficient of variation is more variable.
Long questions with full detailed answers
Q1. Find the mean deviation and standard deviation of the illustrative values 4, 6, 8, 10 and 12, and explain what the result means.
Answer: Introduction: Mean deviation and standard deviation both measure spread, but they treat deviations differently. Mean deviation uses absolute values, while standard deviation squares the deviations, which avoids the problem of signs cancelling out.
Explanation: First find the mean: (4 + 6 + 8 + 10 + 12) / 5 = 40 / 5 = 8. The deviations from the mean are -4, -2, 0, 2 and 4. For mean deviation, take absolute values 4, 2, 0, 2 and 4; their sum is 12, so the mean deviation is 12 / 5 = 2.4. For standard deviation, square the deviations: 16, 4, 0, 4 and 16, whose sum is 40. The variance is 40 / 5 = 8, so the standard deviation is the square root of 8, about 2.83.
Example: If these values were marks in a short quiz, a mean of 8 with a standard deviation of about 2.83 means most marks lie roughly within 2.83 marks of 8, so the class is fairly consistent but not identical.
Conclusion: The mean deviation is 2.4 and the standard deviation is about 2.83. The standard deviation is larger here because squaring gives extra weight to larger deviations, and it is the measure most often used in further statistical work.
Q2. Compare the range, mean deviation and standard deviation, and explain which measure you would choose and why.
Answer: Introduction: All three measures describe spread, but they differ in how much information they use and how sensitive they are to extreme values.
Explanation: The range uses only the largest and smallest values, so it is easy to find but unreliable: one extreme value can change it completely. The mean deviation uses every value and is easy to understand, but ignoring signs by taking absolute values makes it less convenient for further mathematical work. The standard deviation also uses every value, squares deviations so signs do not cancel, and is expressed in the same units as the data, which makes it the most useful measure for comparing sets and for further analysis.
Example: For the illustrative values 5, 6, 7, 8 and 50, the range is 45, which exaggerates the spread because most values are close together, while the standard deviation also rises but reflects the whole set. The quartile deviation would be less affected, but it ignores the middle of the data.
Conclusion: Use the range only for a quick rough idea, use mean deviation for a simple explanation, and use standard deviation for careful comparison and further analysis, because it is the most reliable and mathematically useful measure.
Multiple-choice questions with answer key
- The range of 10, 15, 20 and 30 is:
(a) 10 (b) 20 (c) 30 (d) 5 - The standard deviation is the square root of:
(a) the mean (b) the range (c) the variance (d) the mode - If the variance is 25, the standard deviation is:
(a) 25 (b) 5 (c) 12.5 (d) 625 - The coefficient of variation is found by:
(a) mean / standard deviation x 100 (b) standard deviation / mean x 100 (c) range / mean (d) variance x 100 - Which measure of dispersion is most affected by an extreme value?
(a) quartile deviation (b) range (c) median (d) mode
Answer key: 1-(b), 2-(c), 3-(b), 4-(b), 5-(b).
Chapter 5: Probability
Probability measures how likely an event is to happen. It is always a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. Probability helps us make sensible decisions when the outcome is not known in advance, such as in games of chance, weather forecasts and quality testing.
Important points
An experiment is an action whose result cannot be predicted with certainty, such as tossing a coin. The sample space is the set of all possible outcomes of the experiment. An event is a subset of the sample space, such as getting a head. For equally likely outcomes, the probability of an event is the number of favourable outcomes divided by the total number of possible outcomes. Mutually exclusive events cannot happen together, while independent events do not affect each other’s outcomes. The addition rule is used for the probability of one event or another, and the multiplication rule is used for the probability of one event and another.
Short questions with full detailed answers
Q1. Define probability, and state its limits.
Answer: Probability is a measure of the likelihood that an event will occur. For equally likely outcomes, it is found by dividing the number of favourable outcomes by the total number of possible outcomes. Probability is always between 0 and 1 inclusive. A probability of 0 means the event cannot happen, a probability of 1 means it is certain, and a probability of 0.5 means the event is as likely to happen as not.
Q2. A coin is tossed once. Find the probability of getting a head.
Answer: The sample space for one toss is {head, tail}, so there are 2 possible outcomes. The favourable outcome for getting a head is 1. Therefore the probability is 1 / 2 = 0.5. In the same way, the probability of getting a tail is also 0.5, and the two probabilities add to 1.
Q3. A fair die is rolled once. Find the probability of getting a 4.
Answer: The sample space is {1, 2, 3, 4, 5, 6}, so there are 6 possible outcomes, all equally likely. Only one outcome is favourable, namely 4. Therefore the probability is 1 / 6, which is about 0.167. This shows that any single number is equally likely on a fair die.
Q4. A bag contains 3 red balls and 2 blue balls. A ball is drawn at random. Find the probability that it is red.
Answer: The total number of balls is 3 + 2 = 5. The number of favourable outcomes for drawing a red ball is 3. Therefore the probability is 3 / 5 = 0.6. The probability of drawing a blue ball is 2 / 5 = 0.4, and the two probabilities add to 1, as they must, since every ball is either red or blue.
Q5. Distinguish between mutually exclusive and independent events, with one example of each.
Answer: Mutually exclusive events cannot occur at the same time. For example, on a single roll of a die, getting a 2 and getting a 5 are mutually exclusive, because one roll cannot give both. Independent events are events where the outcome of one does not affect the outcome of the other. For example, tossing a coin twice: getting a head on the first toss does not change the probability of getting a head on the second toss, so the two tosses are independent.
Long questions with full detailed answers
Q1. A fair die is rolled once and a coin is tossed once. Find the probability of getting an even number and a head, and explain the addition and multiplication rules.
Answer: Introduction: This question combines two different rules. The addition rule deals with one event or another, while the multiplication rule deals with one event and another, when the events are independent.
Explanation: For the die, the sample space is {1, 2, 3, 4, 5, 6}, and the even outcomes are 2, 4 and 6, so the probability of an even number is 3 / 6 = 1 / 2. For the coin, the probability of a head is 1 / 2. Because the die and the coin do not affect each other, the events are independent, so the multiplication rule applies: multiply the probabilities, giving (1 / 2) x (1 / 2) = 1 / 4. The addition rule would instead be used for mutually exclusive outcomes of one experiment, such as getting a 2 or a 5 on a single die roll, where the probabilities are added: 1 / 6 + 1 / 6 = 2 / 6 = 1 / 3.
Example: Out of 40 combined trials, we would expect about 10 to give an even number and a head, since 40 x 1 / 4 = 10. This is an illustrative expectation, not a guarantee for any particular set of trials.
Conclusion: The probability of an even number and a head is 1 / 4. Use multiplication for independent events happening together, and addition for mutually exclusive alternatives of the same experiment.
Q2. Explain the basic ideas of probability, with examples of sample space, event, mutually exclusive events and independent events.
Answer: Introduction: Probability is built on a few simple ideas, and once these are clear, most textbook problems become a matter of counting outcomes correctly.
Explanation: An experiment is an action with an uncertain result, such as drawing a card from a pack. The sample space is the set of all possible outcomes; for two coin tosses it is {HH, HT, TH, TT}. An event is a subset of the sample space, such as getting exactly one head, which is {HT, TH}. The probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes. Mutually exclusive events cannot happen together, such as drawing a red card and a black card in a single draw. Independent events do not affect each other, such as two separate tosses of a coin.
Example: For a single die roll, the probability of an even number is 3 / 6 = 1 / 2, because 2, 4 and 6 are favourable out of six outcomes. For two tosses of a fair coin, the probability of two heads is (1 / 2) x (1 / 2) = 1 / 4 by the multiplication rule for independent events.
Conclusion: Probability counts favourable outcomes against all possible outcomes. Knowing whether events are mutually exclusive or independent tells us whether to add or multiply, which is the key to solving probability problems correctly.
Multiple-choice questions with answer key
- The probability of any event lies between:
(a) -1 and 1 (b) 0 and 1 (c) 0 and 2 (d) 1 and 2 - The probability of getting a head on a fair coin is:
(a) 1 (b) 0 (c) 1 / 2 (d) 2 - The sample space for rolling one die has how many outcomes?
(a) 2 (b) 4 (c) 6 (d) 12 - Mutually exclusive events:
(a) always happen together (b) cannot happen together (c) are always independent (d) have probability 1 - For independent events A and B, P(A and B) equals:
(a) P(A) + P(B) (b) P(A) – P(B) (c) P(A) x P(B) (d) P(A) / P(B)
Answer key: 1-(b), 2-(c), 3-(c), 4-(b), 5-(c).
Chapter 6: Index Numbers and Sampling
An index number is a single number that shows the relative change in prices, quantities or values between two periods. The base period is given the value 100, and later periods are compared with it. Sampling means studying a part of the population and using the results to draw conclusions about the whole population, which saves time and cost when studying everyone is not possible.
Important points
A simple price index is found by dividing the current price by the base price and multiplying by 100. A weighted index gives more importance to items that are bought in larger quantities or spent on more heavily. A census studies every member of the population, while a sample studies only a part. Random sampling gives every member an equal chance of selection, which reduces bias. Systematic sampling selects every nth member after a random start, and stratified sampling divides the population into groups called strata and takes a sample from each group, so that every group is represented.
Short questions with full detailed answers
Q1. What is an index number, and what does a value of 120 mean?
Answer: An index number measures the relative change in a variable, such as price, between a base period and a current period. The base period is set at 100. An index of 120 means the variable is 20 percent higher than in the base period. For example, if a price index rises from 100 to 120, prices have risen by 20 percent since the base year.
Q2. A commodity cost 40 rupees in the base year and 50 rupees in the current year. Find the simple price index.
Answer: The simple price index is (current price / base price) x 100. Here it is (50 / 40) x 100 = 1.25 x 100 = 125. This means the price is 25 percent higher than in the base year. The calculation is simple because only one commodity is involved.
Q3. What is the difference between a census and a sample?
Answer: A census collects information from every member of the population, while a sample collects information from only a part of it. A census is more complete but costly, time-consuming and sometimes impossible, for example when the population is very large. A sample is quicker and cheaper, and if it is chosen properly, its results can give a good estimate for the whole population.
Q4. What is random sampling, and why is it used?
Answer: Random sampling is a method in which every member of the population has an equal chance of being selected, often by lottery or by random numbers. It is used because it reduces personal bias in choosing who is included, so the sample is more likely to represent the whole population fairly. Without randomness, the investigator might, even unknowingly, choose members who give a misleading picture.
Q5. Distinguish between systematic sampling and stratified sampling.
Answer: In systematic sampling, members are selected at regular intervals, such as every 10th student on a list, after a random starting point. In stratified sampling, the population is first divided into groups, or strata, such as classes or age groups, and a sample is taken from each group. Stratified sampling is better when groups differ from each other, because it makes sure no important group is left out, while systematic sampling is simpler when a complete list is available.
Long questions with full detailed answers
Q1. Find the simple and weighted price indices for the following illustrative data, and explain the difference between them.
Answer: Introduction: A simple index treats every item equally, while a weighted index gives more weight to items that matter more, so the two can give different answers.
Explanation: Suppose item A costs 20 in the base year and 30 in the current year, with a weight of 2, and item B costs 50 in the base year and 55 in the current year, with a weight of 3. The simple price relatives are (30 / 20) x 100 = 150 for A and (55 / 50) x 100 = 110 for B, so the simple average index is (150 + 110) / 2 = 130. The weighted index weights these relatives: (150 x 2 + 110 x 3) / (2 + 3) = (300 + 330) / 5 = 630 / 5 = 126.
Example: The simple index of 130 overstates the rise because it gives item A, which rose sharply, the same importance as item B, even though item B has the larger weight. The weighted index of 126 is more realistic because it reflects the larger importance of item B.
Conclusion: The simple index is 130 and the weighted index is 126 in this illustrative example. Weighted indices are preferred when items differ in importance, because they reflect real spending or quantities better than a simple average.
Q2. Explain sampling, its advantages and disadvantages, and compare random, systematic and stratified sampling.
Answer: Introduction: Sampling is the study of a part of a population in order to learn about the whole, and the way the sample is chosen decides how far its results can be trusted.
Explanation: Sampling saves time and money, is easier to manage, and can even be more accurate than a rushed census, because a smaller study can be supervised more carefully. Its disadvantages are sampling error, since a sample rarely matches the population exactly, and the risk of bias if the sample is chosen badly. Random sampling gives every member an equal chance and reduces bias, but it needs a complete list and can be awkward in practice. Systematic sampling picks every nth member after a random start, which is simple, but it can go wrong if the list has a hidden pattern. Stratified sampling divides the population into strata and samples from each, which guarantees that every group is represented, but it needs good information about the strata first.
Example: To survey students in a school, a random sample might miss a small class by chance, a systematic sample might pick every 10th name on a list, and a stratified sample would deliberately take students from every class, so no class is left out.
Conclusion: No sampling method is perfect, but a well-chosen sample gives useful results at far lower cost than a census. Stratified sampling is best when groups differ, random sampling is best for fairness, and systematic sampling is best for simplicity when a good list exists.
Multiple-choice questions with answer key
- In the base period, an index number is taken as:
(a) 0 (b) 50 (c) 100 (d) 1000 - A price index of 125 means prices have risen by:
(a) 125 percent (b) 25 percent (c) 12.5 percent (d) 225 percent - A study of every member of a population is called:
(a) a sample (b) a census (c) a stratum (d) an experiment - In random sampling, every member has:
(a) no chance of selection (b) an equal chance of selection (c) a chance based on marks (d) to volunteer - Stratified sampling first divides the population into:
(a) samples (b) strata (c) classes of marks (d) experiments
Answer key: 1-(c), 2-(b), 3-(b), 4-(b), 5-(b).
Revision checklist
Use this checklist before a test. Tick off each point only when you can explain it without looking at the notes.
- Chapter 1: I can define statistics, distinguish a population from a sample, and distinguish primary from secondary data.
- Chapter 2: I can build a frequency distribution, find a class mark, and choose between a bar chart, histogram, frequency polygon and pie chart.
- Chapter 3: I can find the mean, median and mode, and explain when each average is best.
- Chapter 4: I can find the range, variance and standard deviation, and explain what the coefficient of variation is used for.
- Chapter 5: I can find simple probabilities, and I know when to add probabilities and when to multiply them.
- Chapter 6: I can find a simple price index, explain a weighted index, and compare random, systematic and stratified sampling.
- Final check: I have practised one long question from each chapter in writing, and I have checked my MCQs against the answer keys.
Related notes
For other 1st Year material on this site, see the AJK Board 1st Year Arts notes (chapter-wise, all subjects, PDF) here:
https://www.zahidenotes.com/2026/10/ajk-board-1st-year-arts-notes-chapter-wise-all-subjects-pdf.html
For the full 1st Year collection, see the 1st Year hub (11th class notes, guess papers, pairing schemes and past papers) here:
https://www.zahidenotes.com/1st-year-11th-class-notes-guess-papers-pairing-schemes-and-past-papers
