Statistics
In an evenly spaced set, the numbers in the set increase, or decrease, by the same amount and, therefore, share a common difference. If the term "inclusive" is used, it instructs us to include both of the numbers that define the endpoints of the set.
There are 2 ways that evenly spaced sets commonly appear on the GMAT: 1) A set of consecutive (odd or even) integers: {4, 5, 6, 7, 8, 9, 10, 11, 12} 2) A set of consecutive multiples of a given number: {11, 22, 33, 44, 55, 66}
Finding the average (arithmetic mean) of an evenly spaced set 1) Bookend Method: Add the first term to the last term and then divide the sum by 2.
2) The Balance Point Method The average of the set can be visualized as being the midpoint on a line depicting the terms in the set. 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. --> {2, 3, 4, 5, 6}. The average is 4. 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. --> {2, 3, 4, 5, 6, 7}. The average is 4.5.
The average (arithmetic mean) is the sum of all the terms in a set divided by the number of terms in that set
Average = --> Often, we are given a set of terms containing an unknown term (x). The value of which we are required to calculate. X is a term in the set and must be counted with the number of terms
When components of the average are not equally weighted, we have a weighted average. Weighted Average = Sum of Weighted Terms / Total Number of Weighted Terms
Example 23: Two tickets for $ 50 each, 6 tickets for $ 240 each, x tickets for $ 800. Average price $ 260. Weighted average = (q1 x p1) + (q2 x p2) + (q3 + p3) / q1 + q2 + q3 260 = (2 x 50 + 6 x 240 + x x 800) / (8 + x) 540 = 540x x = 1
To compare the standard deviation among data sets that have an equal number of data points, perform the following steps: Step 1: Determine the mean of each set Step 2: For each set, determine the absolute difference between the mean and each data point. Step 3: Sum the differences obtained from each set. The set with the greater sum has the greater standard deviation.
Example: Set A = {2, 5, 6, 7} Set B = {7, 10, 12, 15} Step 1: 20/4 = 5 44/4 = 11 Step 2: 5-2 = 3, 5-5 = 0, 6-5 = 1, 7-5 = 2 11-7 = 4, 11-10 = 1, 12-11= 1, 15-11= 4 Step 3: 3 + 0 + 1+ 2 = 6 4 + 1 + 1 + 4 = 10
Given the value of 2 data points and the quantities of the 2 data points expressed as fractions of the total of items in the weighted average, we can calculate the weighted average of 2 data points.
Example: A room filled with x people. Average shoe size women: 8. Average shoe size men: 12. If 1/4 are women, than 3/4 must be men. WA =
If a set of numbers has n terms and if n is even, the median is the average of the values at n/2 and (n+2)/2 positions, when the numbers are in numerical order.
Example: S = {-8, -1, 3, 5, 6, 9, 10, 11}. n = 8. 8/2 = 4 (8+2) / 2 = 5 The 4th spot is 5 and the 5th spot is 6. The median is (5+6) / 2 = 5.5
If a set of numbers has n terms and n is odd, the median is the vale at the (n+1) / 2 position when the numbers are in numerical order. The formula gives us the position of the median, not the median itself.
Example: S = {-8, -1, 3, 5, 6, 9, 10}. n = 7. (7+1) / 2 = 4 The median is 5.
In any evenly spaced set, the mean of the set is equal to the median of the set.
Example: What are the mean and the median of the first 12 positive multiples of 4? The first 12 positive multiples are between 4 and 48. Mean: (4+48)/2 = 26
To calculate the total number of multiples of A or B, we can use the following formula: The number of multiples of A or B = the number of multiples of A + the number of multiples of B - the number of multiples of LCM(A, B) where LCM(A, B) is the least common multiple of A and B. E.g. How many integers from 1 to 90, inclusive, are multiples of 3 or 4? Number of multiples of 3: (90-3)/3 + 1 = 29 + 1 = 30 Number of multiples of 4: (88-4)/4 + 1 = 21 + 1 = 22 Number of multiples of 12: (84-12)/12 + 1 = 6 + 1 = 7 30 + 22 - 7 = 45
How many integers from 1 to 13, inclusive, are multiples of 2 or 3? --> We must determine the number of elements that are multiples of either of two integers in a set of consecutive integers. Some elements will be multiples of only one of the 2 given integers, while other elements are multiples of both integers at the same time. Therefore, we must be careful not to double-count certain elements. The multiples of 2 are 2, 4, 6, 8, 10, and 12. Thus, we have 6 multiples of 2. The multiples of 3 are 3, 6, 9, and 12. Thus, we have 4 multiples of 3. The true number of multiples is 6 + 4 - 2 = 8. --> We have to remove 2 because 6 and 12 are both multiples of 2 and 3.
The mode is the number that appears most frequently in a data set. 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.
If each number in the data set occurs the same number of times as the others, then there is no mode.
When any set of numbers is placed in numerical order, the median is the value that is in the middle of the arranged set. Half of the numbers above media, other half below median.
If the set has an odd amount of numbers, the median is exactly in the middle of the set. Ex: S = {1, 2, 4, 6, 9, 10, 12}. The median is 6. If the set has an even amount of numbers, the median is the average of the 2 numbers in the middle of the set. Ex: S = {1, 2, 4, 6, 9, 10}. The median is 5, which is the average of 4 and 6.
If we are given that either the largest value or the smallest value in a data set is equal to the mean, all the data points in the set are the same, and thus the standard deviation must be zero
If we have a set of four numbers in which the largest value is 10 and the mean is 10. The values of the other 3 numbers will be: Since sum = average x quantity sum = 10 x 4 = 40 A + B + C + 10 = 40 A + B + C = 30. For the sum of the other numbers to be 30 and for no number to exceed 10, they each must be equal to 10.
The weighted average of 2 different data points will be closer to the data point with the greater number of observations or with the greater weighted percentage.
It is common to be given data points with the weight expressed as a percent. We convert percentages to decimal values and use those as the respective data points. --> Provided that the sum of the percentages is 100% or 1, we can formulate the average as follows: Average = (percent 1) x (data point 1) + (percent 2) x (data point 2) + ... + (percent n) x (data point n) --> If the percentages do not add up to 100%, we must divide by the sum of the percentages we do have. Example: 12% of customers spend $ 10, 18% of customers spend $ 20 0.12($10)+0.18($20) / (0.12+0.18) = $4.80/ 0.3 = $16
Range = Highest Number in a Set - Lowest Number in a Set --> The range is a measure of spread of the data set. The greater the range, the more the numbers are spread out.
More difficult range problems incorporate number properties and algebra. However complex the problems may grow, the range formula does not change.
When considering how to decrease the standard deviation keep in mind that the least possible standard deviation is zero. --> Therefore, the standard deviation must be positive or greater than zero.
One way to guarantee that a positive standard deviation will decrease is to add elements that equal the mean to a set. If you add a number to a set that is close to the mean, the standard deviation will also decrease. However, knowing how close the number must be is very difficult, and thus the GMAT will not test you on that theory.
If we multiply or divide the elements of a data set by a constant amount, the standard deviation will also be multiplied or divided by that amount.
Set A: {$500, $750, $650, $900} --> Average $700 and standard deviation $ 145.8 We multiply each term by 10: Set A: {$5000, $7500, $6500, $9000} --> Average $7000 and standard deviation $ 1458
If we add or subtract the same amount to or from each term in a data set, the standard deviation does not change. That is, we can have data sets with the same standard deviation and different averages.
Set A: {$500, $750, $650, $900} Set B: {$8.000, $8.250, $8.150, $8.400} --> Set B is Set A with $7.500 added to each respective value. The average in set A is $700, but the standard deviation is $145.8. The average in set B is $8.200, but the standard deviation is $145.8.
Maximization/ minimization average problems 40 children won an average of 6 goldfish. 4 children = 3 goldfish. 6 children = 8 goldfish. 5 children = 2 goldfish. 4 no goldfish. Remaining children won at least 4 goldfish. Maximum number of goldfish won by a particular child?
Sum = 6 x 40 = 240 Sum = 240 = 4 x 3 + 6 x 8 + 5 x 2 + 4 x 0 + n n= 170 19 children accounted for. 21 children left for 170 goldfish. 20 children x 4 goldfish per goldfish = 80 170 - 80 = 90 Therefore, one child could win 90 goldfish.
Adding sums Example: The average of a group of 6 students who took an exam was 80. sum = average x # of terms 80 x 6 = 480
The average score for the group of 12 students was 90, what was the average score for the other group of students? SUM(points earned by the first 6 students) + SUM(points earned by the second six students) = SUM(total points earned by all 12 students) --> 480 + 6x = 1.080 --> 6x = 600 --> x = 100
To calculate the number of multiples of A or B, but not of both, we can use the following formula:
The number of multiples of 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)] where LCM(A, B) is the least common multiple of A and B.
To count the number of consecutive integers in a set that includes the first and last numbers, use the formula: (Highest Number - Lowest Number + 1)
To count the number of consecutive multiples of a given number in a range of values that are inclusive of the first and last numbers, use the formula: ((Highest # divisible by the given number - lowest # divisible by the given number)/given number) + 1
To find the number of terms in a set of consecutive integers that includes only one of its endpoints, but not both, subtract the first number from the last number: Last Number - First Number
To find the number of terms between 2 numbers in a set of consecutive integers, subtract the first number from the last number, then subtract 1 from the difference: Last Number - First Number - 1
Average speed (-> rates section) is a specific case of a weighted average --> a "time-weighted" average
We are weighting the speeds based on the time driven at those speeds. As the time for a given speed goes up, the average speed moves closer to that speed. We weight the average by time.
Standard deviation measures "how far" a set of values are from the average (arithmetic mean) of that set. Higher standard deviation: most points in a data set are far from the mean. Lower standard deviation: most points are close to the mean. Zero: all values are the same --> We typically stop at 3 standard deviations. A value greater than 3 standard deviations is almost zero.
We can apply the following formulas to determine a range based on the mean and on the number of standard deviations: High Value = mean + x(sd) Lox Value = mean - x(sd) sd: standard deviation, x: number of standard deviations from the mean Example: One set of math scores for Class C war 86, 79, 90, 92, 76, 81, 83. The mean math score is 87. The standard deviation is 5. How many scores fall within one standard deviation of the mean? Low Value = 87 - 1(5) = 82 High Value = 87 + 1(5) = 92 --> One standard deviation from the mean provides a range of 82 to 92, inclusive. Therefore, 86, 90, 92 and 83 fall within one standard deviation of the mean.
Sometimes, we need to solve average problems in which each of the terms contains a variable.
We use the same formula to consolidate the terms and isolate the variable.
Given the value of 2 data points and the ratio of the quantity of the 2 data points, we can calculate the weighted average of the two data points. Example: 2 Groups of Sharks. Sharks Group No. A: average length of 1.000 cm. Shark Group No. B: average length of 1.600 cm. A = number of sharks in group A. B = number of sharks in group B.
Weighted Average (WA) = {(100 x A) + (1.600 x B)} / A + B --> If we know the ratio between A and B, we'd be able to express one of the variables in terms of the other. A/ B = 2/ 3 --> A = 2/3 B WA = {(100 x 2/3 B) + (1.600 x B)} / {2/3 B + B} WA = 1.360
When the data points of a particular set are not all the same, the standard deviation of that set is greater than zero.
When either the range of a set is not equal to zero or the largest or smallest value in a set is not equal to the mean, the data points are not all the same, and thus the standard deviation is greater than zero.
The standard deviation of a data set will be zero when all the data points in the set are equal to the mean or, in other words, when all the data points are the same
When we are given that either the smallest value of a set is equal to the largest value of a set or the range of a set is equal to zero, we know that all the data points are the same, and thus the standard deviation is equal to zero. Largest value - smallest value = range Largest value - smallest value = 0 Largest value = smallest value
Because all that needs to be known in order to calculate the median is either the middle term, in the case of odd numbered set, or the pair of middle terms, in the case of even numbered sets,
it is sometimes possible to calculate the median even when there are unknown values in the set.
The average formula can be used to find the sum of an evenly spaced set of numbers. Example: What is the sum of the odd integers from 5 to 55, inclusive?
sum = (average)(# of terms) There are (55-5)/2 + 1 = 26 odd numbers in the set. The average is (55+5)/2 = 30. sum = (30)(26) = 780