Statistics GoGMAT, Session 4 I.
Mean or arithmetic average 1) Mean (arithmetic average) of a set of n numbers is the sum of those numbers divided by n. =
.
Example 1: the average of 6, 4, 7, 10, and 4 is
=
= 6.2
2) The mean of the arithmetic progression can be found using the following formula: + × + 2 = = = . 2 So, to calculate the mean of the arithmetic progression there is no need to add all the terms, but only the first and the last. 3) If you are given the average A of a set of n numbers, multiply A by n to get their sum. 4) If all numbers in a set are the same, then that number is the average. 5) If the numbers in the set are not all the same, then the average must be greater than the smallest number and less than the largest number. 6) The sum of the differences between the average and the terms above the average equals the sum of the differences between the average and the terms below the average. Example: Consider the set {83, 84, 86, 98, 99}. Its mean equals 90. The differences between 90 and the terms that are below 90 are 90 – 83 = 7 The total difference is 7 + 6 + 4 = 17. 90 – 84 = 6 90 – 86 = 4 The differences between 90 and the terms that are above 90 are 98 – 90 = 8 The total difference is 8 + 9 = 17. 99 – 90 = 9 These differences are always equal. 7) If there are two groups of objects: Group A
Group B
m objects,
n objects,
mean of x
mean of y The mean for two groups combined: ×
+ +
×
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Example 2: Assume that there are two groups of students: group A consisting of 15 students and group B with 10 students in it. If both groups passed the test and the average scores are 70 and 80 respectively, what is the average score for the two groups combined? To answer the question, use the formula above: 15 × 70 + 10 × 80 1050 + 800 = = 74. 15 + 10 25
8) Note, that if there are two sets A (with average of x) and B (with average of y), then to find the mean for the combined set, you do not actually need to know exact numbers of terms in those sets, but only the ratio of number of terms in A to the number of terms in B. Example: Assume again that there are two groups of students A and B with respective average test scores 70 and 80. If the ratio of number of students in A to the number of student in B is 3:2, then what is the average for the sets A and B combined? To answer the question, use the same formula above, but express the numbers of students as 3s for A and 2s for B: 3 × 70 + 2 × 80 (210 + 160) 370 = = = 74. 3 +2 5 5
Thus, we obtained the same result as in the example above, but without actual values for the numbers of students. 9) If there are two groups of objects, A and B with respective averages m and n, then the average of the two sets combined will be closer to the average of the set with a greater number of objects. Hence, if the numbers of objects are equal, then the average of the two sets combined will be equidistant from the individual averages of the sets A and B. Example 3: In the example 2 we found the average for the two groups with 15 and 10 students respectively, where the individual average scores were given: 70 for A and 80 for B. Here group A has more students than group B, and the average 74 is closer to the average of this group A. Let’s switch the numbers of students: Group A has 10 students with the average score 70 Group B has 15 students with the average score 80. The average for the two groups combined is: 10 × 70 + 15 × 80 700 + 1200 = = 76. 10 + 15 25
76 is closer to 80, the average of the group B with greater number of students. Now assume there are 10 students in both groups: 10 × 70 + 10 × 80 700 + 800 = = 75. 10 + 10 20
75 is equidistant form 70 and 80.
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II.
Median 1) To calculate the median of n numbers, first order the numbers from least to greatest. If n is odd, the median is defined as the middle number If n is even, the median is defined as the average of the two middle numbers. Example 4: In order to find the median of the set {6,4,7,10,4}, put them in order from least to the greatest: {4,4,6,7,10}, and take the central number: 6. Thus the median is 6. For the numbers 4, 6, 6, 8, 9, 12, the median is (6+8)/2=7. 2) Arithmetic progression has its mean equal to its median.
III.
Range 1) The range of the set {x1, x2, x3, ..., xn} is equal to the difference between its greatest and lowest elements.
2) The range is always non-negative:
≥ 0. And it is equal to 0 in two cases:
all the elements are equal to each other {a, a, a…}, the set consists of only one element {a}.
3) Adding the same number to all the terms of the set does not change the range. 4) Multiplying all terms of the set by the same number (m) changes the range. The new range will be m times greater than the initial range.
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IV.
Mode
Mode is a most frequent element in the set. Example: in the set {1, 2, 3, 4, 3, 5, 7, 2, 1, 2} mode is equal to 2. In the set {1, 2, 3, 2, 3} there are two modes, 2 and 3.
V.
Standard deviation 1) If x* is the mean of some finite set S = {x1, ..., xn}, then =
(
−
∗)
+(
−
∗)
+. . . +(
−
∗)
is called standard deviation of S. In fact, you do not need this formula on the GMAT. What you do need is the general (and rough) understanding of the standard deviation as the average of deviations of all the terms from the mean.
mean
(x − x ∗ )
High standard deviation
mean
Low standard deviation
mean
74
2) The standard deviation is always non-negative: equal to 0 in two cases:
3) 4) 5) 6)
≥ 0. And it is
all the elements are equal to each other {a, a, a…}, the set consists of only one element {a}. The standard deviation equals zero if and only if range equals zero. If the same constant a is added to each number of the set {x1, x2, x3,…, xn} with standard deviation s, the deviation of the new set {x1 + a, x2 + a, …, xn + a} will still be s. If each number of the set {x1, x2, x3,…, xn} with standard deviation s is multiplied by the same number b, the deviation of the new set {bx1, bx2, …, bxn} will equal bs. The value of standard deviation is independent from other characteristics, such as mean, median or mode. To obtain the standard deviation we need to know all the values of set members.
Sample problems: 1. Last week, two classes took a test. If the average score for all students in two classes was 82, which class has more students? (1) The average score for the students in class A is 80. (2) The average score for the students in class В is 86. 2. In a certain group of people, the average weight of the males is 170 pounds and of the females, 120 pounds. What is the average weight of the people in the group? (1) The group contains twice as many females as males. (2) The group contains 15 more females than males. 3. If m is the average of the first 10 positive multiples of 5 and if M is the median of the first 10 positive multiples of 5, what is the value of M – m? (A) -5 (B) 0 (C) 5 (D) 25 (E) 27.5
4. For a certain set of n numbers, where n > 1, is the average equal to the median? (1) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2 (2) The range of n numbers in the set is 2×(n – 1) 5.
If the average of four numbers K, 2K + 3, 3K – 5, and 5K + 1 is 63, what is the value of K? (A) 11 (B) 63/4 (C) 22 (D) 23 (E) 253/10
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