Lecture 12 & 13 - Variability

Range

a measure of variability that says how wide the distribution is

Absolute range

tells us the difference between the smallest and largest observed value

Qualities of range

• range can be calculated no matter the shape of the distribution

• finding min and max values is independent of the rest of the distribution

• highly susceptible to outliers

Interquartile range (IQR)

the range of the middle 50% of values; the values between the 25th (Q1) and 75th (Q3) percentile

• tells us how widespread are the observations in the middle of the distribution so it is less affected by outliers (which is good)

For normal distributions, IQR is

symmetrical

For skewed distributions, IQR may be

asymmetric

Box-and-whisker plot

a graph of IQR and range, which helps visualize variability and skew

T/F: Box-and-whisker plots are helpful for comparing distributions

True

Deviation from mean

how spread the data are around the mean

• variability can also be conceptualized as how much each point deviates from the mean

T/F: Greater deviation from the mean signals lesser spread of scores

False

Negative deviation

score less than mean

Positive deviations

score greater than mean

T/F: we can take the average deviation from mean to quantify overall variability

false

T/F: The mean of all deviation scores will always be zero

true

What does mean as a balance point mean?

the points are evenly distributed around the mean

T/F: the total negative deviation will always be the same amount as the total positive distribution

true

How do we quantify deviation from the mean?

one option is to take the absolute value of all deviation scores

Mean absolute deviation (M.A.D)

quantifies the mean of the absolute deviation

• gives a metric for comparing variability

Central tendency in unimodal distributions (numeric order)

• negatively skewed: mean < median < mode

• positively skewed: mode < median < mean

• symmetric: median = mean = mode

Sum of squares

summarizes amount of deviation from mean

Variance

average deviation from the mean in squared units

Degrees of freedom

the number of values that are free to vary in a sample used to estimate a population score

Why is degree of freedom n - 1 for sample variance?

this provides an unbiased estimate of population variance

How does sample size affect the adjustment by n-1?

the bigger the sample, the smaller the effect of adjustment

Population estimate of standard deviation

square root of population variance

Sample standard deviation (SD)

the square root of the sample variance

Standard deviation

average amount by which scores deviate from the mean in original units

When do we use population vs. sample SDs?

• use sample SDs when you measure from the sample to infer about a larger population

• if we can measure the true population mean, we can use population SD

• we typically will use sample SD (s) as we’re wanting to talk about the population

How far is a score from the mean?

we can measure distance from the mean by how many standard deviations away a value is

Interpreting standard deviations

• for most distributions, most values fall within one standard deviation of the mean

• very few data points fall more than 2 SDs away from the mean

Adding variability info to bar graphs

• variability tells us about uncertainty for the mean

• much better to show variability