14.d Standard Deviations, Min/Max

Question 1

What does standard deviation measure?

Answer 1

The dispersion, or spread, of values in a data set from the mean

Question 2

If more data points are farther from the mean, the standard deviation will be _____________

Answer 2

If more data points are farther from the mean, the standard deviation will be LARGER

Question 3

If a stocks mean return was 50%, with a standard deviation of 10%. If the stock’s return is one standard deviation away from the mean, what would it be?

Answer 3

50% - 10% = 40%
50% + 10% = 60%

Two standard deviations away:
50% - 20% = 30%
50% + 20% = 70%

Question 4

What is the formula you would use to determine a range based on the number and number of standard deviations that we wish to be away from it?

Answer 4

High Value = mean + x(sd)
Low Value = mean - x(sd)

Question 5

Is a standard deviation/ mean range inclusive?

Answer 5

Yes!

Question 6

Solve:

Answer 6

One s.d. away range is 82 to 92 (inclusive)

Question 7

How would you set this up?

Answer 7

Question 8

If you add or subtract the same amount to or from EACH TERM in a data set, what happens to the standard deviation?

Answer 8

The standard deviation does NOT change

Question 9

What would happen to the standard deviation if I subtract 1000 from every term in Set B?

Set B: { $8000, $8250, $8150, $8400 }

Answer 9

The standard deviation does NOT change if you subtract or add to ALL terms in a set

Question 10

Solve:

Answer 10

ADDING or SUBTRACTING the same VALUE from every term in a set does NOT change the standard deviation

Question 11

If you MULTIPLY or DIVIDE the same amount to EACH TERM in a data set, what happens to the standard deviation?

Answer 11

The standard deviation will also be MULTIPLIED or DIVIDED by the SAME amount

Question 12

Solve and explain:

Answer 12

1. Dividing by n means you also divide the s.d. by n, DECREASING it
2. Adding n to earn term leaves the s.d. UNCHANGED
3. Multiplying by n means you also MULTIPLY the s.d. by n, INCREASING it

Question 13

What is the least possible standard deviation of a set?

Answer 13

0

Question 14

One way to guarantee that a positive standard deviation will decrease is to add elements that equal the ____________

Answer 14

One way to guarantee that a positive standard deviation will decrease is to add elements that equal the MEAN to a set

Question 15

This set has a mean of 5, and a standard deviation of 1.41,

What happens if you add another 5 to this set?

Answer 15

If you add another 5 (a value that is equal to the mean) to the set, the new standard deviation would become 1.26.

Showing that adding a value of the mean to a set will decrease the standard deviation of it.

Question 16

Solve:

Answer 16

Question 17

How would you compare the standard deviations of data sets that have an equal number of data points?

Answer 17

1. Determine the mean for each set
2. For each set, determine the absolute difference between the mean of the set and each data point in the set (remember: ABSOLUTE)
3. Sum the absolute differences obtained from each individual set

==> The set that has the greatest sum has the largest standard deviation

Question 18

How would you measure which of these sets has the largest standard deviation?

Answer 18

1. Determine the mean:
   Set A = 5, Set B = 11
2. Determine the absolute differences between the mean and each data point in the set (and sum them)
   Set A = 6, Set B = 10

== Therefore Set A must have the smallest standard deviation, and Set B the largest

Question 19

When will the standard deviation of a set equal 0?

Answer 19

If all values in the set are the same

Question 20

What is the standard deviation of: { 4, 4, 4, 4 }?

Answer 20

0

Considering all numbers are the same

Question 21

When the range of a set is ZERO, all data points are the ____________

Answer 21

When the range of a set is ZERO, all data points are the SAME

Question 22

If: Largest Value - Smallest Value = 0, then..?

Answer 22

Largest Value = Smallest Value

and sd must be 0

Question 23

Solve:

Answer 23

Question 24

If the largest, or the smallest value in a data set is equal to the mean….

Answer 24

If the largest, or the smallest value in a data set is equal to the mean….

ALL DATA POINTS ARE THE SAME
(SD must be 0)

Question 25

If the range of a data set is not equal to zero, the data points are NOT all the same, and the standard deviation is ____________ than zero

Answer 25

If the range of a data set is not equal to zero, the data points are NOT all the same, and the standard deviation is GREATER than zero

Question 26

If the largest/ smallest value in a set is not equal to zero, then...

Answer 26

If the largest/ smallest value in a set is not equal to zero, then... NOT all data points are the same, and thus the standard deviation is greater than zero

Question 27

Solve:

Answer 27

Statement 1: only mentions the mean (so this could be 120 on every day and thus sd would be 0). Statement 2: if we know that on one day the number of cars was less than another day, even if we dont know the absolute value we know that Largest - Smallest does NOT equal 0. And thus SD must be greater than 0.

Question 28

Solve:

Answer 28

Key word: distinct

Question 29

Solve:

Answer 29

Question 30

Solve:

Answer 30

Key word: unique