Chapter 4: Variability Flashcards
What is variability?
How spread out the data is
Spread = Variability = Similarity
Low variability =
A: Similar scores
B: Dissimilar scores
A: Similar scores
High variability =
A: Similar scores
B: Dissimilar scores
B: Dissimilar scores
How do we compute variability?
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!
What is variance?
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
What is standard deviation?
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
Which one is the average distance between
the scores and the mean
A. Standard Deviation
B. Variance
C. Sampling Error
A. Standard Deviation
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
True - this is very important to know for this class!
The true standard deviation computed from the entire population of potential participants is the:
A: Parameter
B: Estimate
A: Parameter
The sample standard deviation is an ___________ of the population standard deviation:
A: Parameter
B: Estimate/Approximation
B: Estimate/Approximation
Population standard deviation formula:
𝝈 = square root of ∑ (x - μ) squared ➗ Npop
Population standard deviation = 𝝈 (sigma)
Population = Npop
Sample standard deviation formula:
s = square root of ∑ (x - x̄) squared ➗ N - 1
Sample standard deviation = s
Sample = N-1 (it must always be minus 1)
What is a deviation score?
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
If my score is 20 and the mean is 30, what is
the deviation score?
20 - 30 = -10
True or False: Deviation scores must sum to zero?
True - We can’t sum and average deviation scores
Why do we use squared deviation scores?
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.
What is the sum of squares?
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
True or False: Distances from the sample mean underestimate variation?
True
What are degrees of freedom?
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
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
C. Standard deviation
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
B. Decrease
