## Chi-Square (Χ²) Distributions | Definition & Examples

A chi-square (Χ2) distribution is a continuous probability distribution that is used in many hypothesis tests.

The shape of a chi-square distribution is determined by the parameter k. The graph below shows examples of chi-square distributions with different values of k.

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## Quartiles & Quantiles | Calculation, Definition & Interpretation

Quartiles are three values that split sorted data into four parts, each with an equal number of observations. Quartiles are a type of quantile.

• First quartile: Also known as Q1, or the lower quartile. This is the number halfway between the lowest number and the middle number.
• Second quartile: Also known as Q2, or the median. This is the middle number halfway between the lowest number and the highest number.
• Third quartile: Also known as Q3, or the upper quartile. This is the number halfway between the middle number and the highest number.

Quartiles can also split probability distributions into four parts, each with an equal probability.

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## Pearson Correlation Coefficient (r) | Guide & Examples

The Pearson correlation coefficient (r) is the most common way of measuring a linear correlation. It is a number between –1 and 1 that measures the strength and direction of the relationship between two variables.

Pearson correlation coefficient (r) Correlation type Interpretation Example
Between 0 and 1 Positive correlation When one variable changes, the other variable changes in the same direction. Baby length & weight:

The longer the baby, the heavier their weight.

0 No correlation There is no relationship between the variables. Car price & width of windshield wipers:

The price of a car is not related to the width of its windshield wipers.

Between
0 and –1
Negative correlation When one variable changes, the other variable changes in the opposite direction. Elevation & air pressure:

The higher the elevation, the lower the air pressure.

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## Poisson Distributions | Definition, Formula & Examples

A Poisson distribution is a discrete probability distribution. It gives the probability of an event happening a certain number of times (k) within a given interval of time or space.

The Poisson distribution has only one parameter, λ (lambda), which is the mean number of events. The graph below shows examples of Poisson distributions with different values of λ.

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## Skewness | Definition, Examples & Formula

Skewness is a measure of the asymmetry of a distribution. A distribution is asymmetrical when its left and right side are not mirror images.

A distribution can have right (or positive), left (or negative), or zero skewness. A right-skewed distribution is longer on the right side of its peak, and a left-skewed distribution is longer on the left side of its peak:

You might want to calculate the skewness of a distribution to:

• Describe the distribution of a variable alongside other descriptive statistics
• Determine if a variable is normally distributed. A normal distribution has zero skew and is an assumption of many statistical procedures.

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## Null and Alternative Hypotheses | Definitions & Examples

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test:

• Null hypothesis (H0): There’s no effect in the population.
• Alternative hypothesis (Ha or H1): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable.

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Student’s t table is a reference table that lists critical values of t. Student’s t table is also known as the t table, t-distribution table, t-score table, t-value table, or t-test table.

A critical value of t defines the threshold for significance for certain statistical tests and the upper and lower bounds of confidence intervals for certain estimates. It is most commonly used when:

• Testing whether two means are significantly different (two-sample t tests)
• Testing whether two variables are significantly related (linear regression or correlation)
• Calculating confidence intervals (of means or regression coefficients)

The critical values of t are calculated from Student’s t distribution. Student’s t distribution is the distribution of the test statistic t. The critical values of t are difficult to calculate by hand, which is why most people use a t table or computer software instead.

## Coefficient of Determination (R²) | Calculation & Interpretation

The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome.

Interpreting the coefficient of determination
Coefficient of determination (R2) Interpretation
0 The model does not predict the outcome.
Between 0 and 1 The model partially predicts the outcome.
1 The model perfectly predicts the outcome.

The coefficient of determination is often written as R2, which is pronounced as “r squared.” For simple linear regressions, a lowercase r is usually used instead (r2).

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