Calculating Quartiles, Interquartile Range, and Outliers

Flashcards covering definitions, formulas, and methods for calculating quartiles, interquartile ranges, outliers, deciles, and percentiles from ungrouped data.

Quartile

A value that divides an ordered data set into four equal parts.

Interquartile Range ($IQR$)

The difference between the third quartile ($Q_3$) and the first quartile ($Q_1$) that measures the spread of the middle $50\%$ of the data.

$IQR$ Formula

$$IQR = Q_3 - Q_1$$

Outlier

A value that is unusually high or unusually low compared with the rest of the data. A value that falls outside the expected range may be considered an outlier.

Linear Interpolation

A method used to estimate an unknown value that lies between two known values.

Lower Fence

The lowest acceptable value in a data set; any value below the lower fence is considered an outlier. Formula: $\text{Lower Fence} = Q_1 - 1.5(IQR)$

Upper Fence

The highest acceptable value in a data set; any value above the upper fence is considered an outlier. Formula: $\text{Upper Fence} = Q_3 + 1.5(IQR)$

Steps in Linear Interpolation

1. Find the position of the quartile. 2. Identify the two values between which the quartile lies. 3. Find the difference between the two values. 4. Multiply the difference by the decimal part of the position. 5. Add the result to the lower value. 6. Write the final answer.

Linear Interpolation Usage Note

Linear interpolation is only needed when the position contains a decimal value.

Position of $D_k$

Used to find the $k\text{th}$ decile using the formula: $\frac{k(n + 1)}{10}$

Position of $P_k$

Used to find the $k\text{th}$ percentile using the formula: $\frac{k(n + 1)}{100}$