Range, variance, SD, MAD

What is the range

Shows how spread out the data are by comparing smallest and largest values

Simple measure of variation

Needs interval data

How to calculate the range

1. Identify largest value

2. Identify smallest value

3. Subtract max - min = range

Example of range

Data: 3, 4, 4, 5, 5, 6, 6, 6, 8, 10

Range = 10 − 3 = 7

What is the mean absolute deviation MAD

Shows the average distance of each data point from mean, using absolute values (ignores direction)

How to calculate MAD

1. List the data in x

2. COMPUTE THE MEAN: Add numbers in x column, divide

3. Subtract values from the mean (x-mean)

4. Check, must result in 0

5. Take absolute values

6. Add absolute values

7. Divide sum from absolute value by number of data points

Example of MAD

Data: 3, 7, 6, 4, 5

6/5 = 1.2

MAD requries:

• Interval data

• Can be 0 not negative

• Outliers increase MAD

• Important for SD

What is Variance

Measure of how far data points are from mean but using squared distances

How to calculate Variance

1. Same steps and chart as MAD

2. But square each deviation

3. Add squared values

4. Divide sum by n-1

Example of Variance

Data: 3, 7, 6, 4, 5

Squared sums 10

10/n-1 .. 10/5-1 = 2.5

Why divide by n-1 for variance

Sample variance uses degrees of freedom dividing by n would underestimate population variance

Formula for Standard deviation

Same as Variance, but added square root

What is standard deviation

The square root of variance, most widely used measure of spread

Shows the distance of data points from the mean

How to calculate Standard Deviation

1. Calculate variance

2. Take the square root

Example of Standard Deviation

SD = √Variance

= √2.5

= 1.58

Standard Deviation requires

• Interval or ratio data

• Sensitive to outliers

How are MAD Variance and SD similar?

Same method with the table but for the last column:

• MAD uses absolute values, dividing by n

• Variance squares values, divides by n-1

• SD square root of variance value

What are the differences between SD and MAD

SD

• Squared

• Very sensitive to outliers

• Emphasizes large deviations

• Used in statistics

• Related to normal distribution

MAD

• Absolute deviations

• Somewhat sensitive

• Doesnt emphasizes large deviations

• Rarely used