BISE Lahore 12th Class Statistics Full Book Notes — All Chapters, Questions and Answers

These BISE Lahore 12th class statistics full book notes cover the whole 2nd Year Statistics course for BISE Lahore students: probability, random variables, binomial, Poisson and normal distributions, sampling, estimation, testing of hypotheses, chi-square, and regression and correlation. Each chapter below gives a short summary, the important formulas, short questions with answers, and a long question worked out step by step.

Important note: These are study and revision notes only. They are not official BISE Lahore material, not leaked papers, and they do not guarantee any marks or result. Always check your own textbook and your teacher’s instructions, because the order of chapters and the exact wording of questions can vary by edition.

How to Use These Notes

Use these BISE Lahore 12th class statistics full book notes in four steps. First, read the summary of a chapter and learn its definitions in your own words. Second, copy the formulas onto one revision card and practise them without looking. Third, answer the short questions in writing, then check the answers. Fourth, work the long question slowly, showing every step, because in Statistics most marks are given for method, substitution and a clear conclusion.

• Write the formula first, then substitute the numbers, then simplify, then state the answer with units where needed.
• For distributions, always state the mean and variance before you use a table or a probability.
• For tests, always state the null and alternative hypotheses, the test statistic, the decision rule, and the conclusion in plain words.
• Revise one chapter a day, then mix questions from earlier chapters so you do not forget them.

Quick Revision Before Chapter 1

Statistics in 2nd Year builds on 1st Year ideas. The mean is the average value. The variance measures spread about the mean, and the standard deviation is its square root. A population is the whole group studied, while a sample is a part of it. A parameter describes a population, and a statistic describes a sample. Keep these words exact in short answers, because examiners often ask for the difference between a population and a sample, and between a parameter and a statistic.

Chapter 1: Probability

Summary: Probability measures how likely an event is. An experiment gives outcomes, a sample space lists all possible outcomes, and an event is a set of outcomes. If all outcomes are equally likely, P(A) equals the number of outcomes in A divided by the total number of outcomes. Probabilities always lie between 0 and 1, and the probabilities of all outcomes in a sample space add to 1.

Important formulas: P(A or B) = P(A) + P(B) – P(A and B). For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) x P(B). Conditional probability: P(A given B) = P(A and B) / P(B), provided P(B) is not zero. Complement rule: P(not A) = 1 – P(A).

Short questions: Q1. Define sample space. Ans: The set of all possible outcomes of an experiment. Q2. What is the probability of an impossible event? Ans: 0. Q3. If P(A) = 0.4 and P(B) = 0.3 and A and B are mutually exclusive, find P(A or B). Ans: 0.4 + 0.3 = 0.7.

Long question with steps: A bag has 4 red and 6 blue balls. Two balls are drawn one after another without replacement. Find the probability that both are red. Step 1: P(first red) = 4/10 = 2/5. Step 2: After one red ball is removed, 3 red balls remain out of 9 balls, so P(second red given first red) = 3/9 = 1/3. Step 3: Multiply: (2/5) x (1/3) = 2/15. Answer: 2/15.

Chapter 2: Random Variables and Probability Distributions

Summary: A random variable assigns a number to each outcome of an experiment. A discrete random variable takes separate values, such as the number of heads in three tosses. A continuous random variable can take any value in an interval, such as height. A probability distribution lists the values of a random variable with their probabilities, and the probabilities must add to 1.

Important formulas: Mean, E(X) = sum of x times P(X = x). Variance, Var(X) = E(X squared) – [E(X)] squared. Standard deviation = square root of variance. For a discrete distribution, check first that every probability is between 0 and 1 and that the sum is 1.

Short questions: Q1. What is a discrete random variable? Ans: A variable that takes separate, countable values. Q2. Why must probabilities in a distribution add to 1? Ans: Because one of the listed outcomes must occur. Q3. If E(X) = 5 and E(X squared) = 29, find Var(X). Ans: 29 – 25 = 4.

Long question with steps: X takes values 0, 1, 2 with probabilities 0.2, 0.5 and 0.3. Find E(X) and Var(X). Step 1: E(X) = (0 x 0.2) + (1 x 0.5) + (2 x 0.3) = 0 + 0.5 + 0.6 = 1.1. Step 2: E(X squared) = (0 x 0.2) + (1 x 0.5) + (4 x 0.3) = 0 + 0.5 + 1.2 = 1.7. Step 3: Var(X) = 1.7 – (1.1 squared) = 1.7 – 1.21 = 0.49. Standard deviation = 0.7.

Chapter 3: Binomial Distribution

Summary: The binomial distribution is used when there are a fixed number of independent trials, each trial has only two outcomes, success and failure, and the probability of success stays the same in every trial. Examples are tossing a fair coin a fixed number of times, or checking a fixed number of items for defects.

Important formulas: P(X = r) = C(n, r) x p to the power r x q to the power (n – r), where q = 1 – p. Mean = n x p. Variance = n x p x q. Standard deviation = square root of (n x p x q).

Short questions: Q1. State two conditions for a binomial distribution. Ans: Fixed number of trials, and only two outcomes in each trial. Q2. If n = 10 and p = 0.3, find the mean. Ans: 10 x 0.3 = 3. Q3. What is q if p = 0.25? Ans: 1 – 0.25 = 0.75.

Long question with steps: A fair coin is tossed 4 times. Find P(exactly 2 heads). Step 1: n = 4, p = 1/2, q = 1/2, r = 2. Step 2: C(4, 2) = 6. Step 3: P = 6 x (1/2) squared x (1/2) squared = 6 x 1/16 = 6/16 = 3/8. Answer: 3/8.

Chapter 4: Poisson Distribution

Summary: The Poisson distribution is used for rare events counted in a fixed interval of time or space, such as the number of accidents in a week, or the number of printing errors on a page. It is the limiting form of the binomial distribution when n is large and p is small.

Important formulas: P(X = r) = (e to the power -m) x (m to the power r) / r!, where m is the mean. For a Poisson distribution, mean = variance = m. The standard deviation is the square root of m.

Short questions: Q1. When is the Poisson distribution used? Ans: For rare events in a fixed interval. Q2. If the mean is 4, what is the variance? Ans: 4. Q3. What does m stand for in the Poisson formula? Ans: The mean number of events in the interval.

Long question with steps: On average, 2 errors occur per page. Find P(no error on a page). Step 1: m = 2, r = 0. Step 2: P(X = 0) = (e to the power -2) x (2 to the power 0) / 0! = e to the power -2 x 1 / 1. Step 3: e to the power -2 is about 0.1353. Answer: about 0.1353.

Chapter 5: Normal Distribution

Summary: The normal distribution is a continuous, bell-shaped, symmetric distribution. Its mean, median and mode are equal and lie at the centre. About 68 percent of values lie within one standard deviation of the mean, about 95 percent within two standard deviations, and about 99.7 percent within three standard deviations.

Important formulas: Standard normal variable: Z = (X – mean) / standard deviation. The standard normal distribution has mean 0 and standard deviation 1. To find a probability, convert X to Z, then use the normal table and the symmetry of the curve.

Short questions: Q1. What is the shape of the normal curve? Ans: Bell-shaped and symmetric about the mean. Q2. What are the mean and standard deviation of the standard normal distribution? Ans: Mean 0, standard deviation 1. Q3. What percentage of values lie within two standard deviations of the mean? Ans: About 95 percent.

Long question with steps: Marks are normally distributed with mean 60 and standard deviation 10. Find the Z value for a mark of 75. Step 1: Z = (X – mean) / standard deviation. Step 2: Z = (75 – 60) / 10 = 15 / 10 = 1.5. Step 3: The mark is 1.5 standard deviations above the mean. From the normal table, the area up to Z = 1.5 is about 0.9332, so about 6.68 percent of students score above 75. Answer: Z = 1.5.

Chapter 6: Sampling and Sampling Distributions

Summary: Sampling means studying a part of a population to learn about the whole population. In simple random sampling, every member has an equal chance of being chosen. A sampling distribution is the distribution of a statistic, such as the sample mean, over all possible samples of the same size. The standard error measures how much sample means vary from sample to sample.

Important formulas: Standard error of the sample mean = population standard deviation / square root of sample size, written as sigma / square root of n. If the sample is large, the sampling distribution of the mean is approximately normal, even if the population is not normal. This is the central limit idea students must state in words.

Short questions: Q1. Define a sample. Ans: A part of a population chosen for study. Q2. What is sampling error? Ans: The difference between a sample result and the true population value, caused by studying only a part. Q3. If sigma = 20 and n = 100, find the standard error of the mean. Ans: 20 / 10 = 2.

Long question with steps: A population has mean 50 and standard deviation 12. A sample of 36 is taken. Find the mean and standard error of the sample mean. Step 1: The mean of sample means equals the population mean, 50. Step 2: Standard error = 12 / square root of 36 = 12 / 6 = 2. Step 3: So sample means are centred at 50 with a standard error of 2. Answer: mean 50, standard error 2.

Chapter 7: Estimation

Summary: Estimation uses a sample to guess a population value. A point estimate is a single value, such as a sample mean used to estimate a population mean. An interval estimate, or confidence interval, gives a range that is likely to contain the true value. A 95 percent confidence interval does not mean there is a 95 percent chance for one fixed interval; it means that in repeated sampling, about 95 percent of such intervals would contain the true value.

Important formulas: Confidence interval for a population mean when sigma is known: sample mean plus or minus Z times (sigma / square root of n). For 95 percent confidence, Z is about 1.96. For 90 percent confidence, Z is about 1.645. For 99 percent confidence, Z is about 2.58.

Short questions: Q1. What is a point estimate? Ans: A single sample value used to estimate a population value. Q2. What happens to the interval when the sample size increases? Ans: It becomes narrower, because the standard error falls. Q3. Give the Z value for 95 percent confidence. Ans: About 1.96.

Long question with steps: A sample of 64 students has a mean score of 72, and the population standard deviation is 8. Find a 95 percent confidence interval for the population mean. Step 1: Standard error = 8 / square root of 64 = 8 / 8 = 1. Step 2: Margin = 1.96 x 1 = 1.96. Step 3: Interval = 72 – 1.96 to 72 + 1.96 = 70.04 to 73.96. Answer: about 70.04 to 73.96.

Chapter 8: Testing of Hypotheses

Summary: A hypothesis test checks a claim about a population. The null hypothesis, H0, states no difference or no effect. The alternative hypothesis, H1, states the difference the researcher suspects. A test statistic is calculated from the sample, compared with a critical value, and then H0 is either rejected or not rejected. Not rejecting H0 does not prove it true; it only means the sample does not give enough evidence against it.

Important formulas: For a population mean with sigma known: Z = (sample mean – claimed mean) / (sigma / square root of n). A Type I error is rejecting a true H0. A Type II error is not rejecting a false H0. The level of significance, often 5 percent, is the probability of a Type I error that the test allows.

Short questions: Q1. Define the null hypothesis. Ans: The hypothesis of no difference, assumed true until evidence shows otherwise. Q2. What is a Type I error? Ans: Rejecting H0 when it is true. Q3. If the calculated Z is larger than the critical Z, what is the decision? Ans: Reject H0.

Long question with steps: A claim says the mean is 100. A sample of 25 gives a mean of 104, with sigma = 10. Test at 5 percent, two-sided. Step 1: H0: mean = 100; H1: mean is not 100. Step 2: Z = (104 – 100) / (10 / 5) = 4 / 2 = 2. Step 3: The two-sided critical value at 5 percent is 1.96. Since 2 is greater than 1.96, reject H0. Conclusion: The sample gives evidence that the mean is different from 100.

Chapter 9: Chi-Square Test

Summary: The chi-square test checks whether observed frequencies differ enough from expected frequencies to reject a claim. It is used for goodness of fit, for example whether a die is fair, and for independence in a contingency table, for example whether two attributes are related. The test uses counts, not measurements, and it is always a right-tailed test because squared differences cannot be negative.

Important formulas: Chi-square = sum of [(observed – expected) squared / expected]. Degrees of freedom for goodness of fit = number of categories – 1. For a table with r rows and c columns, degrees of freedom = (r – 1) x (c – 1). Reject H0 if the calculated chi-square is greater than the table value at the chosen level of significance.

Short questions: Q1. What kind of data does the chi-square test use? Ans: Frequencies or counts. Q2. Why is the chi-square test one-sided? Ans: Because squared values are never negative, so only large values give evidence against H0. Q3. A die has 6 faces. What are the degrees of freedom for a goodness-of-fit test? Ans: 6 – 1 = 5.

Long question with steps: A coin is tossed 100 times and gives 60 heads and 40 tails. Test at 5 percent whether the coin is fair. Step 1: H0: the coin is fair. Expected heads = 50, expected tails = 50. Step 2: Chi-square = [(60 – 50) squared / 50] + [(40 – 50) squared / 50] = (100 / 50) + (100 / 50) = 2 + 2 = 4. Step 3: Degrees of freedom = 2 – 1 = 1, and the table value at 5 percent is 3.841. Since 4 is greater than 3.841, reject H0. Conclusion: The coin does not appear fair at the 5 percent level.

Chapter 10: Regression and Correlation

Summary: Correlation measures the strength and direction of a linear relationship between two variables. Regression gives an equation that predicts one variable from the other. A positive correlation means both variables rise together, a negative correlation means one falls as the other rises, and a correlation near zero means no linear relationship. Correlation does not by itself prove that one variable causes the other.

Important formulas: The correlation coefficient, r, lies between -1 and +1. The regression line of Y on X is: Y = a + bX, where b is the slope and a is the intercept. The slope can be found from r times (standard deviation of Y / standard deviation of X). The coefficient of determination, r squared, is the proportion of variation in Y explained by X.

Short questions: Q1. What does r = 0.9 mean? Ans: A strong positive linear relationship. Q2. What does r squared = 0.64 mean? Ans: 64 percent of the variation in Y is explained by X. Q3. Does a high correlation prove causation? Ans: No, correlation alone does not prove causation.

Long question with steps: For two variables, r = 0.8, standard deviation of X = 5, standard deviation of Y = 10, mean of X = 20, mean of Y = 50. Find the regression line of Y on X. Step 1: Slope b = r x (standard deviation of Y / standard deviation of X) = 0.8 x (10 / 5) = 0.8 x 2 = 1.6. Step 2: Intercept a = mean of Y – b x mean of X = 50 – 1.6 x 20 = 50 – 32 = 18. Step 3: Line: Y = 18 + 1.6X. For X = 25, predicted Y = 18 + 40 = 58.

Full-Book Formula Sheet

Probability: P(A or B) = P(A) + P(B) – P(A and B). Independent events: P(A and B) = P(A) x P(B). Mean of a distribution: E(X) = sum of x times P(X = x). Variance: Var(X) = E(X squared) – [E(X)] squared. Binomial: mean = np, variance = npq. Poisson: mean = variance = m. Normal: Z = (X – mean) / standard deviation. Standard error of the mean: sigma / square root of n. Confidence interval: sample mean plus or minus Z times standard error. Test for a mean: Z = (sample mean – claimed mean) / standard error. Chi-square: sum of [(observed – expected) squared / expected]. Regression of Y on X: Y = a + bX.

Final MCQ Revision

Try these 20 MCQs without looking at the answers. The answer key is given below the questions.

1. The probability of a certain event is: (a) 0 (b) 0.5 (c) 1 (d) 2
2. If P(A) = 0.6, P(not A) is: (a) 0.4 (b) 0.6 (c) 1.6 (d) 0
3. A random variable that takes countable values is: (a) continuous (b) discrete (c) constant (d) normal
4. For a probability distribution, the sum of probabilities is: (a) 0 (b) 0.5 (c) 1 (d) 2
5. In a binomial distribution, the number of trials is: (a) random (b) fixed (c) infinite (d) zero
6. If n = 20 and p = 0.5, the binomial mean is: (a) 5 (b) 10 (c) 15 (d) 20
7. In a Poisson distribution, mean and variance are: (a) different (b) equal (c) zero (d) negative
8. The normal curve is symmetric about the: (a) median only (b) mean (c) mode only (d) range
9. The standard normal distribution has mean: (a) 1 (b) 0 (c) 10 (d) 100
10. Z = (X – mean) / standard deviation is used for the: (a) binomial distribution (b) Poisson distribution (c) normal distribution (d) chi-square distribution
11. The standard error of the mean decreases when sample size: (a) decreases (b) increases (c) stays the same (d) becomes zero
12. A 95 percent confidence interval uses Z equal to about: (a) 1.645 (b) 1.96 (c) 2.58 (d) 3.00
13. The null hypothesis states: (a) a difference (b) no difference (c) a cause (d) a correlation
14. Rejecting a true null hypothesis is a: (a) Type II error (b) Type I error (c) correct decision (d) sampling error
15. The chi-square test uses: (a) measurements (b) frequencies (c) ranks only (d) means only
16. Degrees of freedom for a 2 by 3 table are: (a) 1 (b) 2 (c) 3 (d) 6
17. The correlation coefficient lies between: (a) 0 and 1 (b) -1 and +1 (c) 0 and 100 (d) -100 and 100
18. If r = -0.85, the relationship is: (a) weak positive (b) strong negative (c) zero (d) perfect positive
19. r squared = 0.49 means variation explained is: (a) 7 percent (b) 49 percent (c) 70 percent (d) 100 percent
20. A sample is: (a) the whole population (b) a part of the population (c) a parameter (d) a table value

Answer Key: 1-c, 2-a, 3-b, 4-c, 5-b, 6-b, 7-b, 8-b, 9-b, 10-c, 11-b, 12-b, 13-b, 14-b, 15-b, 16-b, 17-b, 18-b, 19-b, 20-b.

Revision Checklist

• I can state the addition, multiplication and complement rules of probability.
• I can find the mean and variance of a discrete distribution.
• I can use the binomial, Poisson and normal distributions and state when each one applies.
• I can find a standard error and a confidence interval for a mean.
• I can set up H0 and H1, calculate a test statistic, and write a conclusion in words.
• I can carry out a chi-square test and interpret a regression line and a correlation coefficient.

Related 2nd Year Posts

• BISE Lahore 12th Class English Compulsory Full Book Notes — All Chapters, Questions and Answers – Book 1 in this 2nd Year set.
• BISE Lahore 12th Class English Compulsory Full Book Notes – the companion full-book revision post for 2nd Year students.

These BISE Lahore 12th class statistics full book notes are for study and revision only. Keep your textbook beside you, practise the worked steps without looking, and check any chapter whose wording differs in your edition with your teacher.