Statistics

Question 1

Mean: Changing by a Constant

Answer 1

• Add/subtract a constant: The mean changes by the same amount added or subtracted to or from each term in a data set
• Multiply/divide a constant: The mean changes by the same amount multiplied or divided by each term in a data set

Question 2

Median: Changing by a Constant

Answer 2

• Add/subtract a constant: The median changes by the same amount added or subtracted to or from each term in a data set
• Multiply/divide a constant: The median changes by the same amount multiplied or divided by each term in a data set

Question 3

Mean Quick Approach:

Limited Set Observation

Answer 3

Mean > SUM(few data) / (# of data points)

• Rationale: Mean = SUM(ALL data) / (# of data points)
• Great method to compare mean and median
• IR application: after sorting the data in ascending order
  → If values in the bottom cells are too high, then Mean > Median
  → If values at the top are too low, and bottom cells are too high, then DON’T apply this approach

Question 4

Mean & Median

Answer 4

Usually, the value of the mean alone isn’t sufficient to calculate the median, UNLESS the set is evenly spaced (in which case median = mean)

In other words, when a set consists of consecutive integers (i.e., evenly spaced), mean = median.

Also, knowing a set’s median and mean is NOT ENOUGH to know the standard deviation.

Question 5

Counting Consecutive Multiples in a Set

Answer 5

To count the number of consecutive multiples of a given number in a given range:

[(Highest # Divisible by the Given Number - Lowest # Divisible by the Given Number) / Given Number] + 1

e.g., Find the number of multiples of 3 between 1 and 100
→ (99 - 3) / 3 + 1 = 33

Question 6

Counting Consecutive Integers:

Inclusive of Either End Points

Answer 6

Note: The following formulas only apply to CONSECUTIVE integers (e.g., 1, 2, 3, 4, 5) specifically, NOT any evenly-spaced set of integers. BUT the same concept applies to non-consecutive but evenly spaced integers (just use the Counting Consecutive Integers formula).

• Inclusive of both end points: Last Number - First Number + 1
• Inclusive of only 1 end point: Last Number - First Number
• Excluding both end points: Last Number - First Number - 1

Question 7

Ways to Find the Mean

Answer 7

(These methods are only applicable when the set is EVENLY spaced)

Bookend Method: When we have an evenly spaced set of numbers
→ Avg = (1st + Last) / 2

Balance Point Method:

• In an evenly spaced set with an odd number of terms, the average of the terms in the set is the exact middle term of the set when the terms are in numerical order.
• In an evenly spaced set with an even number of terms, the average of the terms in the set is the average of the two middle terms of the set when the terms are in numerical order.

Question 8

Counting the Multiples of Integer A or B in a Set of Consecutive Integers

Answer 8

To calculate the total number of multiples A or B:

The number of multiples of A + the number of multiples of B - the number of multiples of LCM(A, B)

Question 9

Counting the Multiples of Integer A or B, but not of both, in a Set of Consecutive Integers

Answer 9

To calculate the total number of multiples A or B, but not of both:

The number of multiples of A + the number of multiples of B - 2 x the number of multiples of LCM(A, B)

Question 10

Weighted Averages

Answer 10

The weighted average of 2 data points can be calculated if the following #1 + #2 are known:

1. The value of the 2 data points
2. The quantities of the 2 data points
   → Way #1: ratio of the 2 quantities
   → Way #2: fractions or percentages of the total number of items

If we know the values of both the 2 data points and the total average value, we know the ratio between the quantities of the 2 data points, but NOT the actual quantities of each.

Question 11

Median: Finding the Position

Answer 11

Odd: If a set of numbers has n terms and n is odd, the median is the value at the (n+1) / 2 position when the numbers are in numerical order.

Even: If a set of numbers has n terms and n is even, the median is the average of the values at the n / 2 and (n + 2) / 2 positions when the numbers are in numerical order.

Question 12

Median: Unknown Numbers

Answer 12

Sometimes, but not always, the median can be determined even when there are unknown values in a set of numbers.

Method: strategically test numbers

Question 13

When Mean = Median

Answer 13

Mean = Median anytime there is an EVENLY spaced set of numbers

Question 14

Mode

Answer 14

• More than 1 mode: If two (or more) numbers appearing in a data set occur more frequently than others and these numbers occur the same number of times, then all of these numbers are modes of the data set.
• No mode: If each number in the data set occurs the same number of times as the others, then there is no mode.

Question 15

Range

Answer 15

• Maximize range: Increase the largest number or decrease the smallest number
• Minimize range: Decrease the largest number or increase the smallest number

Question 16

Range = 0

Answer 16

Range = 0 when the largest number of a set is equal to the smallest number of a set
→ This means all data points are the same
→ Which means SD = 0

When range = 0, SD = 0

Question 17

Range: when the difference of 2 numbers is known

Answer 17

1 is sufficient.

e.g., If S is a set of four numbers w, x, y, and z, is the range of the numbers in S greater than 2 ?

1. w − z > 2
2. z is the least number in S

Question 18

Standard Deviation: Changing by a Constant

Answer 18

• Add/Subtract by a constant: SD doesn’t change when we add or subtract the same amount to or from each term in a data set. We can have data sets with the same SD and different averages.
• Multiply/Divided by a constant: SD also multiplies/divides by that constant.
• GMAT won’t ask you to calculate SD

Question 19

Standard Deviation and Mean

Answer 19

• The least possible SD of a set is 0
• A positive SD will decrease when elements equal to the mean are added to a set (to be exact, elements that are close to the mean, but this concept won’t be tested)

Question 20

Comparing Standard Deviation without Fully Calculating It

Answer 20

To compare the SD among data sets that have an equal number of data points, perform the following steps:

1. Determine the mean of each set
2. For each individual set, determine the absolute difference between the mean of that set and each data point in that set
3. Sum the differences obtained from each individual set

The set with the greater sum has the greater SD

Question 21

Standard Deviation = 0

Answer 21

• SD = 0 when all of the values in a data set are the same
• SD > 0 when the values in a data set are not the same

Note: Mean = Median DOESN’T mean SD = 0

Question 22

Clues that data points in a set are all the same

Answer 22

1. Range = 0
2. SD = 0
3. When the largest or smallest value is equal to the mean

Also means that when any of these 3 scenarios are not true, the data points aren’t all equal, and SD > 0.

Question 23

Testing scenarios to find the median

Answer 23

When the question provides a list of numbers, with two of them unknown (e.g., x and y), calculate the max median and min median possible to understand the range of possibilities.