JEE Main

Mean, Median, Mode, and Range

• Mean (Average): Sum of all values divided by the number of values.
• Median: Middle value when the data is arranged in ascending order.
• Mode: Value that appears most frequently.
• Range: Difference between the highest and lowest values.

Probability of an Event

• Ratio of the number of favorable outcomes to the total number of possible outcomes.
• Can be expressed as a fraction, decimal, or percentage.

Conditional Probability

• Probability of an event occurring given that another event has already occurred.
• Written as P(A | B), where A is the event of interest and B is the condition.

Bayes’ Theorem

• Used to calculate the probability of an event occurring based on prior knowledge and conditional probabilities.
• P(A | B) = (P(B | A) * P(A)) / P(B)
• P(B | A) = conditional probability of event B occurring given that event A has occurred
• P(A) = probability of event A occurring
• P(B) = probability of event B occurring

Binomial Distribution

• Discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability p.
• P(X = x) = (nCx) * p^x * q^(n-x)
• nCx = n! / (x!(n-x)!)
• p = probability of success
• q = probability of failure = 1 - p
• n = number of independent experiments

Normal Distribution

• Also known as the Gaussian distribution or bell curve.
• A continuous probability distribution that is symmetric around the mean with a bell-shaped curve.
• Completely determined by its mean and standard deviation.

Standard Deviation and Variance

• Standard deviation (σ): Square root of the variance.
• Variance (σ^2): Average of the squared differences between each value in the data set and the mean.
• Measures the spread or dispersion of data.
• A low standard deviation indicates that the data is clustered close to the mean, while a high standard deviation indicates that the data is spread out.

Correlation and Regression

• Correlation: Measures the strength and direction of a linear relationship between two variables.
• Linear regression: Models the relationship between a dependent variable and one or more independent variables using a linear equation.

CBSE Class 11-12

Mean, Median, Mode, and Range

• Same as JEE Main.

Probability of an Event

• Same as JEE Main.

Conditional Probability

• Same as JEE Main.

Binomial Distribution

• Same as JEE Main.

Normal Distribution

• Same as JEE Main.

Standard Deviation and Variance

• Same as JEE Main.

Correlation and Regression

• Same as JEE Main.

Frequency Distributions

• Tally marks or a table to organize data into intervals and record the frequency of each interval.

Cumulative Frequency Distributions

• Sum the frequencies of all intervals up to and including a given interval.

Histograms

• Graphical representation of a frequency distribution using bars.

Measures of Central Tendency

• Mean: Average value.
• Median: Middle value when the data is arranged in ascending order.
• Mode: Value that appears most frequently.

Measures of Dispersion

• Range: Difference between the highest and lowest values.
• Standard deviation: Square root of the variance.
• Variance: Average of the squared differences between each value in the data set and the mean.

Skewness and Kurtosis

• Skewness: Measures the asymmetry of a data distribution.
• Kurtosis: Measures the peakedness or flatness of a data distribution.

Sampling Techniques

• Techniques used to select a sample from a population to represent the entire population.
• Simple random sampling: Each individual in the population has an equal chance of being selected.
• Stratified sampling: Population is divided into strata, and a sample is taken from each stratum.
• Cluster sampling: Population is divided into clusters, and a sample of clusters is taken.

Hypothesis Testing

• Statistical method used to determine whether a hypothesis is supported or refuted based on sample data.
• Null hypothesis (H₀): Statement being tested.
• Alternative hypothesis (H₁): Statement that contradicts the null hypothesis.
• P-value: Probability of obtaining a test statistic as extreme as or more extreme than the observed test statistic, assuming the null hypothesis is true.
• Significance level (α): Maximum p-value at which the null hypothesis can be rejected.

Linear Regression

• Used to model the relationship between a dependent variable and one or more independent variables using a linear equation.
• Equation: y = mx + b
• y: dependent variable
• x: independent variable
• m: slope
• b: y-intercept