Assume all percentages are decimals unless stated otherwise.

2 Descriptive Statistics

Percentiles

Where $k$ is the percentile, $i$ the index (ranking position) of the value in question, and $n$ the total number of values,

\[\begin{align} i &= k(n+1) \end{align}\]

Inner Quartile Range

\[\begin{align} \text{IQR} &= Q_3 - Q_1 \end{align}\]

Determining Outliers through IQR

An data point is considered an outlier if it is outside the IQR by an amount greater than 1.5 times the IQR.

\[\begin{align} [𝑄_1−1.5⋅\text{IQR},𝑄_3+1.5⋅\text{IQR}] \end{align}\]

Mean of a Frequency Table

Sum your midpoints multiplied by their frequency, then divide by the total frequency.

\[\begin{align} \mu &= \frac{\sum fm}{\sum f} \end{align}\]

Z-scores

Where $x$ is the value, $\mu$ the mean, and $\sigma$ the standard deviation,

\[\begin{align} z &= \frac{x - \mu}{\sigma} \end{align}\]

Standard Deviation

Where $x$ represent each data value, $\mu$ the mean, and $n$ the total number of values,

\[\begin{align} \sigma &= \sqrt{\frac{\sum (x - \mu)^2}{n}} \end{align}\]

In other words,

  1. Find the differences between each data point and the mean
  2. Square each one
  3. Add them all up
  4. Divide by the number of data points
  5. Square root that quotient

3 Probability Topics

Mutual Exclusion

To find the probability of event $A$ given that event $B$ has already happened,

\[\begin{align} P(A | B) &= \frac{P(A \text{ and } B)}{P(B)} \end{align}\]

Basic Rules of probability

The multiplication rule for AND events.

\[\begin{align} P(A \text{ and } B) &= P(A|B)\cdot P(B) \\ \end{align}\]

The addition rule for OR events.

\[\begin{align} P(A \text{ or } B) &= P(A) + P(B) - P(A \text{ and } B) \end{align}\]

4 Discrete Random Values

Expected Value or Mean of a Discrete Probability Distribution

Where $x$ is the value and $P(x)$ the probability of that value occurring, sum all the products of the two.

\[\begin{align} \mu &= \sum xP(x) \end{align}\]

Standard Deviation of a Discrete Probability Distribution

Where $x$ is the value, $P(x)$ the probability of that value occurring, and $\mu$ the mean,

\[\begin{align} \sigma &= \sqrt{\sum P(x)(x-\mu)^2 } \end{align}\]
  1. Find the differences between each data point and the mean
  2. Square each one
  3. Multiply the squares by the probability of each data value occurring
  4. Add them all up
  5. Square root that quotient

5 Continuous Random Variables

Mean of a Uniform Distribution

Where $a$ and $b$ are the smallest and largest values,

\[\begin{align} \mu &= \frac{a + b}{2} \end{align}\]

Standard Deviation of a Uniform Distribution

Where $a$ and $b$ are the smallest and largest values,

\[\begin{align} \sigma &= \sqrt{\frac{(b-a)^2}{12}} \end{align}\]

7 Central Limit Theorem

Standard Deviation of a Sampling Distribution

Also known as the standard error of the mean. Use this as your standard deviation when when finding z-scores of sampling distributions. $\sigma$ is the standard deviation of the variable and $n$ the sample size.

\[\begin{align} \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \end{align}\]