Notes from Toppers
Measures of Central Tendency:
 Mean (Arithmetic Mean)
 Definition as sum of observations divided by the number of observations. (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Mean of Grouped Data)
 Median:
 Definition as middle value when the observations are arranged in ascending order (Refer to NCERT Class 11, Chapter 15, Statistics (Part 1)  Median of Ungrouped Data)
 Mode:
 Definition as the value that appears most frequently in the data (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Mode of Ungrouped Data)
Measures of Dispersion:
 Range:
 Defined as the difference between the largest and smallest observations (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Range of Grouped Data)
 Variance:
 Definition as the sum of squared deviations from the mean divided by number of observations minus one (Refer NCERT Class 12, Chapter 25, Probability)
 Standard Deviation (SD):
 Defined as square root of the variance. Represents how spread out the observations are from the mean (Refer NCERT Class 12, Chapter 25, Probability)
 Quartile Deviation (Q.D or H):
 Defined as half of the difference between upper and lower quartiles. (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Quartile Deviation)
 Interquartile Range (IQR):
 Defined as the difference between the upper and lower quartiles. (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Interquartile Range)
Skewness and Kurtosis:
 Skewness:
 Describes the asymmetry in the data distribution, can be positive or negative (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Karl Pearson’s Coefficient of Skewness)
 Kurtosis:
 Describes the peakedness or flatness of the data distribution compared to a normal distribution (Refer NCERT Class 11, Chapter 15, Statistics (Part 1)  Karl Pearson’s Coefficient of Kurtosis)
Correlation and Regression:
 Scatter Diagrams:
 A graphical representation that shows the relationship between two variables where each data point represents a pair of measurements. (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Scatter Plot)
 Karl Pearson’s Coefficient of Correlation (r):
 A measure of the linear relationship between two variables, ranges from 1 to 1. (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Correlation)
 Spearman’s Rank Correlation Coefficient (r_{s}):
 Used to measure the monotonic relationship between two variables. (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Spearman’s Rank Correlation)
 Regression Analysis:
 Used to predict the value of one variable (dependent variable) based on the values of one or more other variables (independent variables).
 Linear Regression Equation:
 An equation in the form y = mx + b, where ’m’ is the slope and ‘b’ is the intercept. Commonly used for linear regression.
 Residuals:
 The difference between observed values and predicted values in regression analysis. (Refer NCERT Class 12, Chapter 25, Probability)
Probability:
 Basic Concepts (Sample Space, Events, Probability) (Refer NCERT Class 12, Chapter 13, Probability)
 Conditional Probability:
 The probability of occurrence of an event given that another event has already occurred. (Refer NCERT Class 12, Chapter 13, Probability)
 Bayes’ Theorem:
 A method to calculate conditional probabilities based on prior probabilities, likelihoods, and total probabilities. (Refer NCERT Class 12, Chapter 13, Probability)
 Independent Events:
 Events are said to be independent if the occurrence of one event does not affect the probability of occurrence of the other.
 Mutually Exclusive Events:
 Events are mutually exclusive if the occurrence of one event precludes the occurrence of the other.
 Addition Rule of Probability:
 If (A) and (B) are two events, the probability of either (A) or (B) occurring is (P(A \cup B) = P(A) + P(B)  P(A \cap B)).
 Multiplication Rule of Probability:
 If (A) and (B) are two events, the probability of both (A) and (B) occurring is (P(A \cap B) = P(A) \cdot P(BA)).
Random Variables and Probability Distributions:
 Random Variable:
 A variable whose value is determined by the outcome of a random event.
 Discrete Probability Distributions:
 Probability distribution of discrete random variables, commonly Binomial, Poisson and Hypergeometric distributions (Refer NCERT Class 12, Chapter 13, Probability)
 Continuous Probability Distributions:
 Probability distribution of continuous random variables, commonly including the normal and exponential distributions. (Refer NCERT Class 12, Chapter 13, Probability)
 Central Limit Theorem:
 States that the distribution of sample means approaches the normal distribution as sample size increases, regardless of the shape of the population distribution.
Sampling Techniques :
 Simple Random Sampling: Every member of the population has an equal chance of getting selected. (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Random Sampling)
 Stratified Random Sampling: The population is divided into groups/strata and then simple random sampling is carried out within each stratum (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Stratified Random Sampling)
 Cluster Random Sampling: Instead of selecting individual elements randomly, groups or clusters of individuals are randomly selected from the population (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Cluster Sampling)
 Systematic Random Sampling: Selecting individuals at regular intervals from a predetermined starting point (Refer NCERT Class 11, Chapter 15, Statistics (Part 2)  Systematic Sampling)
Hypothesis Testing:
 Null Hypothesis (H_{0}) and Alternative Hypothesis (H_{1}):
 H_{0} is a statement that there is no significant difference between two populations, while H_{1} suggests the opposite. (Refer NCERT Class 12, Chapter 15, Probability)
 Type I and Type II Errors:
 Type I Error (Rejecting H_{0} when it’s true) and Type II Error (Accepting H_{0} when it’s false).
 Level of Significance:
 The probability at which the null hypothesis is rejected when it is actually true.
 Critical Value:
 The value that separates the rejection region from the acceptance region in hypothesis testing.
 Pvalue:
 The probability of obtaining a test statistic as extreme as or more extreme than the observed sample result, assuming that the null hypothesis is true. (Refer NCERT Class 12, Chapter 15, Probability)
 Onesample Ztest:
 Used to test whether the sample mean is equal to a specified value when population variance is known.
 Twosample Ztest:
 Used to test whether the means of two independent normally distributed populations are equal when population variances are known.
 Chisquare test:
 Used to determine if there is a significant difference between observed frequencies and expected frequencies of categories. (Refer NCERT Class 12, Chapter 15, Probability)
 Student’s ttest:
 Used when population variance is unknown, to test whether the mean of a population is equal to a specified value and to compare the means of two populations.
 Oneway ANOVA:
 Used to compare means of more than two independent groups/populations.
Confidence Intervals:
 Confidence Interval for Mean:
 Defined as a range of possible values that contains the true population mean with a certain confidence level. (Refer NCERT Class 12, Chapter 15, Probability)
 Confidence Interval for Proportion: (Refer NCERT Class 12, Chapter 15, Probability)
 Confidence Interval for Variance:
 Defined as a range of values that contains the true variance of the population with a specified level of confidence.
 Confidence Interval for Regression Slope:
 Defined as a range of values that contains the true slope of the regression line with a specific confidence level.
NonParametric Tests:
 Sign Test:
 Used to compare two matched samples when data is not normally distributed. (Refer NCERT Class 12, Chapter 14, Mathematical Reasoning)

Wilcoxon Signedrank Test: