Statistics
What is the sum of integers from 101 to 202, inclusive?
sum = (average)(number of terms) number of terms = 202 - 101 +1 = 102 average = (202 - 101) / 2 = (303/2) sum = (303/2)*(102) = 15,453
What is the sum of the odd integers from 5 to 55, inclusive?
sum = (average)*(# of terms) # of terms = ((55-5)/2) +1 = 50/2 + 1 = 26 odd #'s in the set average = (55+5) / 2 = 60/2 = 30 sum = 30*26 = 780
What is the sum of the odd integers from 400 to 999, inclusive?
sum = average x quantity avg = (999 + 401) / 2 = 1400/2 = 700 qty = (999 - 401) / 2 + 1 = 299 + 1 = 300 sum = (700)*(300) = 210000
On the first even day of the month, Steve saves $5, and on each following even day he saves $5 more than he saved the previous even day of the month. If there were 30 days in the month, how much money did Steve save by the end of the month?
sum = average x quantity quantity = (Last Day # - First Day Number)/2 + 1 = 28/2 +1 = 15 average for a progression = first integer + last integer / 2 5 + 75 / 2 = 40 sum = average x quantity = $40 x 15 = $600
What is the sum of every one-digit and two-digit multiple of 5?
sum = avg x qty avg = (first + last) / 2 = (5 + 95) / 2 = 100/2 = 50 qty = (last - first) / 5 + 1 = (95-5)/5 + 1 = 90/5 + 1 = 18 + 1 = 19 sum = 50 x 19 = 950
What does "inclusive" mean when defining the endpoints of a set of numbers?
"Inclusive" instructs us to include both of the numbers that define the endpoints of the set, the first and last numbers of the set.
What equation would you use to find the number of multiples in a given set? For example: find the number of multiples of 3 between 1 and 100, inclusive.
((highest # divisible - lowest # divisible) / given number) +1 ((99-3)/3) + 1 = 33
A survey was conducted to determine how much money 95 shoppers spent on holiday gifts. 15 shoppers spent $50 each, 20 shoppers spent $72 each, 10 shoppers spent $80 each, 12 shoppers spent $95 each, and 38 shoppers spent $100 each. Of the 95 shoppers surveyed, what was the median number of dollars spent on holiday gifts?
(95 + 1) / 2 = 96/2 = 48 (the 48th number will be the median) $50 1-15 $72 16-35 $80 36-45 $95 46-57 $100 58-95 $95 is the median
How many numbers between 100 and 500, inclusive, are odd and multiples of 4?
0 = all multiples of 4 will be even
In a group of college freshmen and sophomores, do more than 15% of the students in the group use laptops? 1) 15% of sophomores use laptops 2) 10% of freshmen use laptops
1 and 2 alone are not sufficient, we need more info about both 1&2) SUFFICIENT - the average will be between 10 and 15% no matter what, the quantity does not matter, but it will not be >15%
How do you compare the standard deviation among data sets with an equal number of data points?
1) Determine the average 2) Determine the absolute difference between the mean and each data point in the set 3) Sum the differences from each difference
A marketing company tested 6 different ovens. The cooking times for the first three ovens were a, b, and c minutes. These three cooking times produced a standard deviation of d minutes. Was the standard deviation of the cooking times of the next three ovens greater than d minutes? 1) The average for the last three ovens was equal to the average of the first three ovens. 2) The cooking times for the last three ovens was equal to a + x, b + x, and c + x minutes.
1) Have the same average, does not mean the range is the same. The standard deviation could be smaller or larger. 2) If we add a constant value to each piece of data in a set, the standard deviation does not change, so the standard deviation for the last three ovens will also be d minutes.
At a certain annual dance tournament, was the number of competitors in rounds 1 to 3 greater than the number of competitors in rounds 3 to 5? 1) The average number of competitors in rounds 4 and 5 was twice the number of competitors in rounds 1 and 2. 2) There were 20 contestants in rounds 1 and 2.
1) Since round 3 is included in both rounds, we only need to focus on round 1 and 2 and 4 and 5. (r4+r5) / 2 = 2(r1 + r2) r4 + r5 = 4(r1 + r2) r4 + r5 > r1 + r2 SUFFICIENT 2) NS
Over a span of 10 baseball games, what was the standard deviation of the number of hits made by Derek? 1) The mean number of hits made by Derek was 3. 2) The median number of hits made by Derek was 3.
1) avg = 3 NS 2) med = 3 NS 1&2) we do not know the range, therefore we do not know the standard deviation NS None of the statements are sufficient.
A team of automotive machines have produced an average of 10 cars per day during the last 10 days. For how many days would the machines have to average 20 cars per day to raise the overall average to 12 cars per day? (How would you setup this equation?)
12 = (10*10 + 20x) / (10 + x) 10*10 = 100 cars (10 cars/day in 10 days) 20x = 20 cars per day for x days 10 + x = there are 10 days initially and we are adding a new number of x days
Five consecutive integers are added together and sum to 115. What is the result when 27! is divided by the product of these five integers?
5 consecutive integers: x + (x +1) + (x+2) + (x+3) + (x+4) + (x+5) = 5x + 10 5x + 10 = 115 x = 21 ===> 21, 22, 23, 24, 25 27! = 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20! / (21 x 22 x 23 x 24 x 25) = 27 x 26 x 20!
What is the median of the following numbers: (see back side)
= 0 ; = 1/3 ; = ~1.7/3; = ~2.2/3; = ~2.6/3 sqrt(3)/3 is the median
How can you determine the average of an evenly spaced set of terms? (Bookend Method)
Add First Number + Last Number / 2
A = 12, 15, 20, 21, 10 B = 12+x, 15+x, 20+x, 21+x, 10+x If Set A has a standard deviation of n, which of the following is the standard deviation of Set B?
Because Set B is Set A with a constant term (x) added to each term in the set, the standard deviation does not actually change. Therefore, the resulting standard deviation of Set B is still n.
The table represents the number of hot dogs eaten at a contest and the number who ate the number of hot dogs. What was the median number of hot dogs eaten in the contest? # of people hot dogs eaten 40 5 22 10 20 15 10 20
For large data sets where it's difficult to order the numbers, it is more efficient to find the total number of terms and then calculate which term is in the middle of the set. There are 40 + 22 + 29 + 10 = 101 terms in the data set. (101 + 1) / 2 = 51 This means the median is in the 22 # of people row, so 10 hot dogs is the median.
What is the formula for the number of consecutive integers in a set that includes the first and last numbers?
Highest Number - Lowest Number + 1
How would you describe Standard Deviation?
How far a set of values are from the average
What is the sum of all two-digit numbers that leave a remainder of 1 when divided by both 3 and 4?
If a number leaves a remainder of 1 when divided by both 3 and 4, it must leave a remainder of 1 when divided by 12, since 3 x 4 = 12. The number pattern is 12n+1 => sum = avg x qty Since this is a pattern (or arithmetic progression): => avg = (first + last) / 2 = (13 + 97) / 2 = 110/2 = 55 => qty = ((last - first))/12 + 1 = (97-13)/12 + 1 = 84/12 + 1 = 8 sum = avg x qty = 55 x 8 = 440
The average of a set of five unique positive integers is 17. If the median of this set is 20, what is the maximum possible value of the largest integer in the set? Also, what does it mean if a set contains five unique positive integers?
If a set contains, five unique positive integers, all of the numbers in the set, must be different (but positive integers). avg = sum / 5 17 = (a + b + c + d + e) / 5 85 = a + b + 20 + d + e 65 = a + b + d + e Now we must minimize the values. a and b are less than 20 and d must be greater than 20. 65 = 1 + 2 + 21 + e e = 41
How do you find the number of terms in a set of consecutive integers that includes only one of its endpoints, but not both?
Last Number - First Number
How do you find the number of terms in a set of consecutive integers that exclude the endpoints?
Last Number - First Number - 1
What is the standard deviation if the range is 0?
The standard deviation is also 0 as there would not be numbers higher or lower than the average.
What happens to the standard deviation if you multiple or divide the elements of a data set by a constant amount?
The standard deviation will also be multiplied or divided by that amount.
What is the range of a set?
Range = highest # in a set - lowest # in a set
How do you calculate the median, if a set of numbers has n terms and n is even and the median is in between two sets of numbers?
Take the average of the two numbers
How can you determine the average of an evenly spaced set of terms? (Balance Point Method)
The average is the exact middle term of the set. Ex: 2, 3, 4, 5, 6 4 is the average
What happens if you have a set of evenly spaced numbers in terms of the mean and median?
The mean is equal to the median
What is the median of a set?
The middle value of a set when the set is numerically ordered.
What is the Mode of a set of numbers?
The mode is the number that appears most frequently in a data set.
The sum of any odd number of consecutive integers will always be divisible by what?
The number of the odd consecutive integers.
What is an evenly spaced set?
The numbers in the set increase or decrease by the same amount and therefore share a common difference.
What is the mode of a data set?
The value that appears most often.
The average of 8 consecutive integers is 20.5. What is the average of the first 7 of these integers?
There are two ways to solve this: 1) let x represent the first variable and x + 7 represent the 8th variable 20.5 = (x + x + 7) /2 2x + 7 = 41 x = 17 avg = (17 + 18 + 19 + 20 + 21 + 22 + 23) /7 = 20 2) 20.5 is the middle of the set, so the range is: 17 18 19 20 (20.5) 21 22 23 24 remove the last term: 17 18 19 20 21 22 23 = the average is going to be the middle of the set, so the average is 20
What must be true about 5 consecutive positive integers? I. The mean is equal to the median II. The mean is greater than the median III. The product of the 5 integers is evenly divisible by 5 IV. The sum of the 5 integers is evenly divisible by 5
We can let the 5 numbers be x, x+1, x+2, x+3, x+4 1. True - for consecutive integers, the mean will be equal to the median avg = (5x+10)/5 = x + 2 med = x + 2 2. False - The mean will equal the median, it cannot be greater 3. True - The product of any 5 consecutive integers is always divisible by 5, because one of the 5 numbers must have a 5 or a 0 as its last digit 4. True - since we already divided the sum by 5 for the average, we know this is true Rule: The sum of any odd number of consecutive integers will always be divisible by that number.
What is the equation for weighted averages?
Weighted average = (sum of weighted terms)/(total # of weighted terms)
When X is divided by 8, the remainder is 4. If 0 < X < 100, what is the sum of all values of X?
X/8 = Q + 4/8 X = 8Q + 4 sum = avg x qty first = 8(0) + 4 = 4 last = 8(11) + 4 = 92 avg = (92 + 4) / 2 = 96/2 = 48 qty = (92 - 4) / 8 + 1 = 11 + 1 = 12 sum = 48 x 12 = 576
What is the equation for average (arithmetic mean)?
average = sum of terms / number of terms
How do you determine the high/low values for standard deviation?
high value = mean + x(sd) low value = mean - x(sd) where x is the number of standard deviations from the mean
What is the equation for average of several groups? (Note this can be extended to 3 or more groups)
n1*a1 + n2*a2 = na set has n numbers with an average of a group 1 has n1 numbers with an average of a1 group 2 has n2 numbers with an average of a2
If we know the average of score of 6 students (group A) is 80 and if the average of 12 students (group B) is 90, what is the average score for the other group of 6 students? (how would you write this equation)?
sum of A + sum of B = total sum points earned by all 12 students sum of A = 6*80 = 480 total sum = 12*90 = 1080 480 + 6x = 1080 6x = 600 x = 100
What is the equation for the sum of terms?
sum of terms = average * # of terms