Standard Deviation

Show your work for each problem.

1.
What does standard deviation measure?
1. a)The difference between the largest and smallest values
2. b)The average of the data
3. c)How spread out the data is from the mean
4. d)The largest value in the data
Answer: How spread out the data is from the mean
Standard deviation measures how spread out the data is from the mean. A small standard deviation means data is clustered near the mean; a large one means data is spread far from the mean.
2.
If a data set has a standard deviation of , what can you conclude?
1. a)The mean is zero
2. b)The data contains negative numbers
3. c)All values are the same
4. d)There are no values in the data
Answer: All values are the same
A standard deviation of means there is no spread at all - every value is exactly the same as the mean, which means all values are identical.
3.
If the variance of a data set is , what is the standard deviation?
Answer: 4
. The standard deviation is always the square root of the variance.
4.
If the standard deviation is , what is the variance?
Answer: 9
Variance . Since standard deviation is the square root of variance, variance is the square of standard deviation.
5.
Data set A: Data set B: Both have mean . Which has the larger standard deviation?
1. a)Data set A
2. b)Data set B
3. c)They have the same standard deviation
4. d)Cannot be determined
Answer: Data set B
Data set A has because all values equal the mean. Data set B has values spread from to , so it has a larger standard deviation (actually ).
6.
Find the mean of:
Answer: 6
Mean
7.
Calculate the standard deviation of: (Mean = )
Answer: 1.63
1. What is ? 4
2. What is ? 0
3. What is ? 4
4. What is the variance (mean of squared deviations)? 2.67
5. What is (rounded to 2 decimals)? 1.63
8.
Stock A has returns with . Stock B has . Which stock is riskier?
1. a)They have the same risk
2. b)Stock A
3. c)Stock B
4. d)Cannot determine from this information
Answer: Stock B
Stock B is riskier because its higher standard deviation ( vs ) means its returns vary more - you could gain or lose more money.
9.
For the data , calculate the standard deviation.
Answer: 0
1. What is the mean of ? 4
2. What is for each value? 0
3. What is the variance? 0
4. What is ? 0
10.
Data: (Mean = ). What is the sum of squared deviations?
Answer: 40
11.
Calculate the standard deviation of: (Mean = )
Answer: 2.83
1. What is the sum of squared deviations? 40
2. What is the variance ()? 8
3. What is (rounded to 2 decimals)? 2.83
12.
If you add to every value in a data set, what happens to the standard deviation?
1. a)It stays the same
2. b)It becomes
3. c)It doubles
4. d)It increases by
Answer: It stays the same
Adding the same value to all data points shifts the entire distribution but doesn't change the spread. The mean increases by , but every deviation stays the same, so the standard deviation is unchanged.
13.
If you multiply every value in a data set by , what happens to the standard deviation?
1. a)It doubles
2. b)It stays the same
3. c)It quadruples
4. d)It is halved
Answer: It doubles
Multiplying all values by doubles both the mean and each deviation. Since standard deviation is based on deviations, it also doubles. (Variance would quadruple because it's based on squared deviations.)
14.
Calculate the standard deviation: (Mean = )
Answer: 2.83
1. What are the deviations from the mean? -4, -2, 0, 2, 4
2. What are the squared deviations? 16, 4, 0, 4, 16
3. What is the sum of squared deviations? 40
4. What is the variance? 8
5. What is the standard deviation (2 decimals)? 2.83
15.
Factory A makes bolts with mm. Factory B makes bolts with mm. How many times more consistent is Factory A?
Answer: 4
. Factory A's bolts have 4 times less variation, making them 4 times more consistent than Factory B's.