Statistics

Question 1

there are many problem variations that can occur where you need to use the arithmetic mean formula where you may be given the mean, and then a variable is inserted into the (sum of set of elements)/(# of elements) part

Question 1

arithmetic mean?

Answer 1

sum of set of elements/ # of elements

Question 2

how do you deal with situations where you are total group is broken up and you are given information about averages of sub components of the group and about the group as a whole and asked for average information about the other subcomponent

Answer 2

Question 3

what is an evenly spaced set

Answer 3

numbers in the set increase or decrease by the same amount, and thus, share a common difference

Question 4

how do you count the number of consecutive integers in a set (inclusive of the first and last elements)?

Answer 4

highest number - lowest number +1
ex: how many numbers from 50 to 101 inclusive? 101-50+1=52

Question 5

how do you count # of consecutive multiples in a set, inclusive?

Answer 5

((highest # divisible by the multiple - lowest # divisible by the multiple)/multiple number)+1
ex: number of multiples of 3 between 1 and 100 inclusive is (100-1)/3 + 1

Question 6

how do you count the number of consecutive items in a set (including only one of the end points of the set)?

Answer 6

simply subtract smallest number in the set by the biggest number

Question 7

how do you count the number of consecutive integers between the first and last numbers in a set

Answer 7

biggest - smallest - 1

Question 8

what is the bookend method for finding the arithmetic mean?

Answer 8

if you have a set of evenly spaced numbers, the arithmetic mean is (largest+smallest)/2

Question 9

what is the balance point method for finding the arithmetic mean?

Answer 9

for set with even number of terms: balance point method means you find the two middlemost terms and take their average
for set with odd number of terms: average of the set is the middlemost point

Question 10

how do you calculate the number of multiples of A or B in a set of consecutive integers?

Answer 10

multiples of A + # multiples of B - # multiples of LCM(A,B)

LCM(A,B) must be removed since those numbers will be duplicated in each list and we want them to only appear once

Question 11

how do you calculate the number of multiples of A or B BUT NOT OF BOTH in a set of consecutive integers?

Answer 11

[multiples of A] + [# multiples of B] - 2*[# multiples of LCM(A,B)]
similar to calculating the number of multiples of A or B, but you must remove all instances of the LCM(A,B) by removing it twice

Question 12

weighted average = ?

Answer 12

((data pt 1)(frequency of data pt 1)+(data pt 2)(frequency of data pt 2) + ….)/(total frequency of data points)

Question 13

what assertions can be made about a weighted average when there are differences in the frequency of a certain data point?

Answer 13

the weighted average of two different data points will be closer to the data point with the greater frequency, number of observations, or weighted percentage

Question 14

how can you still calculate weighted averages if all you have is the data points and the ratio of the prevalence of the two groups?

Answer 14

given the value of two data points and the ratio of the quantity of the two data points, we can calculate the weighted average

Question 15

what is the median of a set of numbers?

Answer 15

if set has an odd number of items: it is the number that is exactly in the middle after the set has been ordered
if set has an even number of items: order the set of numbers, then take the average of the two middle numbers - that is the median

Question 16

how do you quickly determine the POSITION of the median of a large ordered set with an odd number of items?

Answer 16

for a set with n items (where n is odd), the median is in the (n+1)/2 position of the ordered set

Question 17

how do you quickly determine the POSITION of the median of a large ordered set with an even number of items?

Answer 17

for a set with n items (where n is even), the median the average of the values at the (n+2)/2 and n/2 positions of the ordered set

Question 18

when are the mean and median of a set of numbers the same?

Answer 18

if the set of numbers is equally spaced

Question 19

what is the mode?

Answer 19

the number that occurs most frequently in a set

Question 20

can a set have more than one mode?

Answer 20

yes, if two or more numbers appear more frequently in a dateset than other numbers they are all modes

Question 21

can a dataset have no mode?

Answer 21

yes, if all numbers occur with the same frequency

Question 22

What is the range of a dataset?

Answer 22

[highest value in set] - [lowest value in set]

Question 23

what does standard deviation measure?

Answer 23

how far a set of values are from the arithmetic mean of that data set

Question 24

establishing ranges based on the mean and how many standard deviations from the mean you want to be:

Answer 24

high value = mean + [x # of st dev][st dev]
low value = mean - [x # of st dev][st dev]

Question 25

does the standard deviation change between two datasets if one dataset is simply the original dataset plus some constant value added to each term?

Answer 25

no, in this case the mean would change but the actual standard deviation wouldn’t change cause the center of gravity of the distribution simply shifted, the spread between points did not

Question 26

does the standard deviation change between two datasets if one of the datasets is simply the original dataset multiplied or divided by some constant value?

Answer 26

yes, the st dev changes by the factor of that constant value

Question 27

fact:

Answer 27

if we multiply or divide the elements of a dataset by a constant amount, the standard deviation will also be multiplied or divided by that constant amount

Question 28

what is the minimum possible standard deviation of a set?

Answer 28

zero

Question 29

what is one thing you can do to a set of numbers with a positive (non zero) standard deviation to decrease the standard deviation of the set?

Answer 29

add elements to the set that are equal to the mean of the set

Question 30

How to compare standard deviations of data set with an equal number of data points?

Answer 30

Step 1: Determine the mean of each set
Step 2: For each set, determine the absolute difference between the mean of the sets and each individual point in the sets (sum of differences between mean of each set and points in each set)
Step 3: Sum the differences in each set
Step 4: The set with the greatest sum of differences has the greatest st dev

Question 31

when will the standard deviation of a set be zero

Answer 31

when all the points in the set are the same

Question 32

if the smallest and largest values of a set are the same, what does that say about the range and st dev of the set?

Answer 32

range = 0 -> all values in set are the same -> st dev = 0

Question 33

what would it mean if the largest or smallest value of a dataset is equal to the mean of the data set?

Answer 33

this means that all the points in the dataset are the same and thus the range =0 and st dev = 0 .. proof: a set of 4 numbers has a mean of 10 and the largest value of the set is 10. since ave = sum/#, sum = ave# = 104 = 40. so let the four values sum to 40 -> 10+a+b+c=40, so a+b+c=30. since no number can be greater than 10, all of a, b, and c must equal 10 for the equality to hold given the restriction

Question 34

when can you be certain that the st dev of a data set is not zero (and that it is some value greater than zero)

Answer 34

when not all the values of the set are equal

Question 35

what criteria are such that if any of the three are true, the st dev is greater than zero?

Answer 35

1) the range is not zero
2) the smallest and largest numbers in the set are not equal
3) the smallest or largest number in the set is not equal to the mean of the set