### Shortcut Methods

**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