Calculating Standard Deviation

Show your work for each problem.

1.
What is the FIRST step when calculating standard deviation?
1. a)Square each value
2. b)Take the square root
3. c)Find each deviation
4. d)Find the mean (average)
Answer: Find the mean (average)
The first step is always finding the mean. You need the average to calculate how far each value deviates from it.
2.
Why do we square the deviations in the standard deviation formula?
1. a)To make the calculation easier
2. b)Because that's how the formula works
3. c)So negative and positive deviations don't cancel out
4. d)To get larger numbers
Answer: So negative and positive deviations don't cancel out
Squaring makes all deviations positive. Without squaring, and would cancel to , hiding the actual spread of the data.
3.
Find the mean of these values:
Answer: 8
Mean
4.
If the mean is , what is the deviation for the value ?
Answer: -3
Deviation . The negative sign shows that is below the mean.
5.
What is ?
Answer: 16
. Negative times negative equals positive!
6.
Which data set has a standard deviation of ZERO?
1. a)
2. b)
3. c)
4. d)
Answer:
When all values are identical (), every deviation is , making the variance and SD .
7.
Data: . Mean = . Sum of squared deviations = ?
Answer: 8
Deviations: , , . Squared: . Sum .
8.
When do you divide by instead of ?
1. a)When your data is a sample from a larger population
2. b)When calculating variance
3. c)When you have more than 10 values
4. d)When the mean is negative
Answer: When your data is a sample from a larger population
Use for sample SD when your data represents only a portion of the entire population. This correction gives a better estimate of the true population SD.
9.
Calculate the variance for: (mean = )
Answer: 2.67
1. What is the deviation for ? -2
2. What is the deviation for ? 2
3. What are the squared deviations? (as a sum) 8
4. Variance = sum ÷ 3 = ? (round to 2 decimals) 2.67
10.
Find the standard deviation:
Answer: 8.16
1. What is the mean? 20
2. List the deviations -10, 0, 10
3. Sum of squared deviations? 200
4. Variance = 200 ÷ 3 ≈ ? 66.67
5. SD = √66.67 ≈ ? 8.16
11.
Calculate the population standard deviation:
Answer: 1.41
1. What is the mean? 6
2. What is the deviation for ? -2
3. What is the deviation for ? 2
4. Sum of ALL squared deviations? 10
5. Variance = 10 ÷ 5 = ? 2
6. SD = √2 ≈ ? 1.41
12.
If the variance is , what is the standard deviation?
Answer: 5
SD . The relationship: , so .
13.
You add to every value in a data set. What happens to the standard deviation?
1. a)It doubles
2. b)It decreases
3. c)It stays the same
4. d)It increases by
Answer: It stays the same
Adding a constant shifts the mean and all values by the same amount, but the distances (deviations) between values stay the same. SD measures spread, not location, so it's unchanged.
14.
Calculate the SAMPLE standard deviation for: (use )
Answer: 3
1. What is the mean? 15
2. List the deviations -3, 0, 3
3. Sum of squared deviations? 18
4. Sample variance = 18 ÷ (3-1) = ? 9
5. Sample SD = √9 = ? 3
15.
Quiz scores: . Calculate the population standard deviation.
Answer: 7.07
1. What is the mean? 80
2. List the 5 deviations -10, -5, 0, 5, 10
3. Sum of squared deviations? 250
4. Variance = 250 ÷ 5 = ? 50
5. SD = √50 ≈ ? 7.07