Calculate SS, variance, and standard deviation for a population.
Calculate SS, variance, and standard deviation for a sample.
Tools You Will Need
Summation notation (Chapter 1)
Central tendency (Chapter 3)
Mean
Median
4-1 Introduction to Variability
Variability can be defined as:
A quantitative distance measure based on the differences between scores.
Describes the spread of scores or distance of a score from the mean.
Purposes of measure of variability:
Describe the distribution (clustered vs. spread out).
Measures how well an individual score represents the distribution.
Why Study Variability?
Variability helps to differentiate between distributions, even if they have similar means.
Example: Two treatments (A and B) might have different means, but the variability within each treatment group can influence the interpretation of the results.
Three Measures of Variability
Range
Variance
Standard deviation
Range
Discrete Variable: Highest score – Lowest score.
Example: 1, 2, 3, 4, 5, 15. Range = 15 – 1 = 14.
Continuous Variable: (Upper Real Limit of Highest score) – (Lower Real Limit of Lowest score).
4-2 Defining Variance and Standard Deviation (1 of 4)
Standard deviation:
Most common and important measure of variability.
A measure of the standard or average distance from the mean.
Describes whether the scores are clustered closely around the mean or are widely scattered.
Calculation differs for population and samples.
Variance:
A necessary companion concept to standard deviation.
Symbols for Population
Mean = \mu
Sum of Squares = SS
Variance = \sigma^2
Standard Deviation = \sigma
Simple Steps for Standard Deviation (Population)
Step 1: Sum of Squares (SS). {SS = \sum(X - \mu)^2}
Step 2: Variance (\sigma^2) = {SS \div N}
Step 3: Standard Deviation (\sigma) = \sqrt{\sigma^2}
Sum Of Squares
The formula for sum of squares is: {SS = \sum X^2 - {(\sum X)^2 \over N}}
4-4 Measuring Variance and Standard Deviation for a Sample
Goal of inferential statistics:
Draw general conclusions about the population based on limited information from a sample.
Samples differ from the population:
Samples have less variability.
Computing the variance and standard deviation in the same way as for a population would give a biased estimate of the population values.
The bias in sample variability is consistent and predictable, which means it can be corrected.
Symbols for Sample
Mean = M
Sum of Squares = SS
Variance = s^2
Standard Deviation = s
Simple Steps for Standard Deviation (Sample)
Step 1: Sum of Squares (SS). {SS = \sum(X - M)^2}
Step 2: Variance (s^2) = {SS \over (n - 1)}
Step 3: Standard Deviation (s) = \sqrt{s^2}
Why (n - 1)?
To fix the bias in sample variability, we use degrees of freedom (df), which is the number of scores in the sample that are independent and free to vary.
Corrects for the fact that we used the sample mean.
When using your sample mean to calculate spread:
You're not using the true average from the whole population, so the variation looks smaller than it really is.
Dividing by (n – 1) instead of n corrects this and gives a better estimate of the real (population) spread.
Degrees of Freedom
Population: Mean is known, so degrees of freedom = N.
Sample: Mean is unknown, so degrees of freedom = n – 1.
Sample vs Population formula
Step 1: Sum of Squares (same)
Step 2: Variance (different = for sample we use n – 1 or df instead of N)
Step 3: Standard Deviations (Same)
Presenting the Mean and Standard Deviation in a Frequency Distribution Graph
For both populations and samples, it is easy to represent mean and standard deviation:
A vertical line in the “center” denotes the location of the mean.
A horizontal line to the right, left (or both) denotes the distance of one standard deviation.
Transformations of Scale
Adding a constant to each score:
The mean is changed.
The standard deviation is unchanged.
Multiplying each score by a constant:
The mean is changed.
The standard deviation is also changed and is multiplied by that constant.