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(B|A)).

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 (H₀) and Alternative Hypothesis (H₁):
  • H₀ is a statement that there is no significant difference between two populations, while H₁ suggests the opposite. (Refer NCERT Class 12, Chapter 15, Probability)
• Type I and Type II Errors:
  • Type I Error (Rejecting H₀ when it’s true) and Type II Error (Accepting H₀ 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.
• P-value:
  • 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)
• One-sample Z-test:
  • Used to test whether the sample mean is equal to a specified value when population variance is known.
• Two-sample Z-test:
  • Used to test whether the means of two independent normally distributed populations are equal when population variances are known.
• Chi-square 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 t-test:
  • 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.
• One-way 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.

Non-Parametric Tests:

• Sign Test:
  • Used to compare two matched samples when data is not normally distributed. (Refer NCERT Class 12, Chapter 14, Mathematical Reasoning)

• Wilcoxon Signed-rank Test: