chad 2

Range

Measures the spread of a dataset by subtracting the smallest value from the largest.

Formula for Range

Range = Maximum - Minimum.

Example of Range

For data 3, 7, 8, 12, 15, Range = 15 - 3 = 12.

Key Ideas about Range

Simple measure of spread, uses only two values (min and max), sensitive to outliers.

Quartiles

Divide a dataset into four equal parts.

Q1 (First Quartile)

The value below which 25% of the data falls.

Q2 (Median)

The value below which 50% of the data falls.

Q3 (Third Quartile)

The value below which 75% of the data falls.

Steps to Find Quartiles

1. Order the data, 2. Find the median (Q2), 3. Find Q1 and Q3.

Interquartile Range (IQR)

IQR = Q3 - Q1.

Key Ideas about Quartiles

Describes the middle 50% of data and is less affected by outliers.

Standard Deviation

Measures how far data values typically are from the mean.

Population Formula for Standard Deviation

$ext{σ} = \frac{ ext{Σ}(x - ext{μ})^2}{N}$.

Sample Formula for Standard Deviation

$s = \frac{ ext{Σ}(x - \bar{x})^2}{n-1}$.

Steps to Calculate Standard Deviation

1. Find the mean, 2. Subtract mean from each value, 3. Square differences, 4. Find average of squared differences, 5. Take the square root.

Key Ideas about Standard Deviation

Small standard deviation indicates data is close to the mean; large standard deviation indicates data is spread out.

Percentiles

Indicate the percentage of data values at or below a certain value.

Example of Percentiles

The 70th percentile means 70% of data is at or below that value.

Common Percentiles

25th percentile = Q1, 50th percentile = Median (Q2), 75th percentile = Q3.

Steps to Find a Percentile

1. Order the data, 2. Use percentile formula to find position, 3. Identify or interpolate the value.

Key Ideas about Percentiles

Shows relative standing within a dataset; useful for comparing individuals.

Summary of Range

Uses min and max, tells total spread, sensitive to outliers.

Summary of Quartiles (IQR)

Uses middle 50%, tells central spread, not sensitive to outliers.

Summary of Standard Deviation

Uses all data, tells typical distance from mean, sensitive to outliers.

Summary of Percentiles

Uses position in data, tells relative standing, not sensitive to outliers.

When to Use Range

For a quick overview of spread.

When to Use Quartiles/IQR

When data has outliers or is skewed.

When to Use Standard Deviation

When working with mean and normal distributions.

When to Use Percentiles

To compare position within a dataset (e.g., test scores).