Chapter 4: Variability

Question 1

What is variability?

Answer 1

How spread out the data is

Spread = Variability = Similarity

Question 2

Low variability =

A: Similar scores
B: Dissimilar scores

Answer 2

A: Similar scores

Question 3

High variability =

A: Similar scores
B: Dissimilar scores

Answer 3

B: Dissimilar scores

Question 4

How do we compute variability?

Answer 4

Step 1: Determine each score’s distance from the center of the data (usually mean)

Step 2: Averaging the distances

Basically, you’re using the mean to compute variability!

Question 5

What is variance?

Answer 5

The variance is the average SQUARED distance
between the scores and the mean.

s squared = ∑ (x - x̄) squared ➗ N - 1

In other words, it’s the sum of squares ➗ the adjusted sample size (df)

(df) = degrees of freedom

The adjusted sample size in the denominator —
the degrees of freedom — improves accuracy

The variance isn’t a useful descriptive tool
because it’s hard to think about squared things

Question 6

What is standard deviation?

Answer 6

The standard deviation is the average distance
between the scores and the mean.

It’s intuitive because its value is on the same scale as the data.

s = square root of ∑ (x - x̄) squared ➗ N - 1 = the square root of s squared

The sample standard deviation approximates the
population standard deviation, the average
distance to the mean in the population

Question 7

Which one is the average distance between
the scores and the mean

A. Standard Deviation
B. Variance
C. Sampling Error

Answer 7

A. Standard Deviation

Question 8

True or False:

68% of the scores in a normal distribution fall
within ± 1 standard deviation of the mean, and
~ 95% are within ± 2 standard deviations

Answer 8

True - this is very important to know for this class!

Question 9

The true standard deviation computed from the entire population of potential participants is the:

A: Parameter
B: Estimate

Answer 9

A: Parameter

Question 10

The sample standard deviation is an ___________ of the population standard deviation:

A: Parameter
B: Estimate/Approximation

Answer 10

B: Estimate/Approximation

Question 11

Population standard deviation formula:

Answer 11

𝝈 = square root of ∑ (x - μ) squared ➗ Npop

Population standard deviation = 𝝈 (sigma)
Population = Npop

Question 12

Sample standard deviation formula:

Answer 12

s = square root of ∑ (x - x̄) squared ➗ N - 1

Sample standard deviation = s
Sample = N-1 (it must always be minus 1)

Question 13

What is a deviation score?

Answer 13

A deviation score is the distance between a
score and the sample mean (center of the data)

x - x̄ = distance

X = specific score (specific person)
x̄ = sample mean

Deviation scores can be positive (score > mean)
* Score = 33 / Mean = 30 / 33-30 = Deviation of +3

Deviation scores can be negative (score < mean)
* Score = 28 / Mean = 30 / 28-30 = Deviation of -2

Deviation scores can be zero (score = mean)
* Score = 30 / Mean = 30 / 30-30 = Deviation of 0

Question 14

If my score is 20 and the mean is 30, what is
the deviation score?

Answer 14

20 - 30 = -10

Question 15

True or False: Deviation scores must sum to zero?

Answer 15

True - We can’t sum and average deviation scores

Question 16

Why do we use squared deviation scores?

Answer 16

We eliminate negative values by working with
squared distances from the mean

(x - x̄) squared = squared distance from center

You’ll do this after you compute deviation scores. Now everything will be positive and won’t cancel out and equal zero.

Question 17

What is the sum of squares?

Answer 17

The sum of squares is the sum of the squared distances from the mean OR the sum of squared distances between a set of scores and the mean (the numerator of the variance and standard deviation)

∑ (x - x̄) squared

The sum of squares is the building block for measures of variability.

Keep in mind that we don’t know the population mean. Substituting the sample mean makes the sum of squares too small, dividing by N − 1 compensates

s squared = ∑ (x - x̄) squared ➗ N - 1

S squared = sample variance
𝝈 squared = population variance

Question 18

True or False: Distances from the sample mean underestimate variation?

Answer 18

True

Question 19

What are degrees of freedom?

Answer 19

The degrees of freedom can be viewed as an
adjusted sample size

The sum of squares in the numerator is too
small, dividing by N − 1 counteracts this bias

Question 20

The average deviation between the scores and the sample mean is:

A. Deviation score
B. Sampling error
C. Standard deviation
D. Variance
E. Sum of Squares

Answer 20

C. Standard deviation

Question 21

Imagine a negatively skewed histogram with scores between 10 and 40. If we remove the scale scores with values between 10 to 15 from the dataset, what will happen to the standard deviation?

A. Increase
B. Decrease
C. Keep the same

Answer 21

B. Decrease