Find a set of numbers that will satisfy the following conditions:
- the median of a set of 20 numbers is 24.
- the range is 42.
- to the nearest whole number the mean is 24.
- no more than three numbers are the same.
Start with your scores at the mean/median. Spread them out from there, with extreme scores at 3 and 45 (mean/median ± 21).
I found this answer on another site, it is accurate. I had this same problem for my stats class.
The easiest way to do this would be to construct the set based on the conditions. So start from the middle and work out.
The median of the 20 numbers is 24. The median is the middle point and since there is two middle points (the tenth and eleventh number) it's the average of the two. But to make it easy just make both the tenth and eleventh number 24.
Since the average (mean) of the numbers is 24, if you add the same amount to the right as you subtract from the left, you'll always maintain the average of 24. So add or subtract (1) to each side.
on the right side you would have {...24, 25, 26,27,28,29, 30 31, 32, 33}
on the left you would have {15, 16, 17, 18, 19, 20, 21, 22, 23, 24...}
this would give the set {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26,17,18,19, 30 31, 32, 33}
now, this set fits three of the four condition, but the range (the largest - the smallest) is not 42, its 18 ((33-15=18.)
but that's easy enough to find. We want to find your minimum (min) and your maximum (max) with the following conditions :
1. max - min = 42
2. 24 - x = min
3. 24 + x = max
substituting 3 and 2 back into 1, you get the following equation
(24 + x) - (24 - x) = 42
24 + x - 24 + x = 42
24 - 24 + 2x = 42
2x = 42
x = 21
therefore, your max is (24 +21)= 45 and your min is (24 - 21) = 3
replace those max and min with the max and min from the previous set and you get an answser.
{3,16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26,27,28,29, 30 31, 32, 45}
To satisfy the given conditions, we can follow these steps:
Step 1: Determine the range
Since the range is given as 42, it means the difference between the maximum and minimum numbers in the set should be 42.
Step 2: Determine the maximum value
Since the mean is approximately 24, and 20 numbers are in the set, the sum of all numbers in the set should be (24 x 20) = 480. Since the maximum value should be as close as possible to the mean to not affect the mean much, we can assume the maximum value to be 24.
Step 3: Determine the minimum value
Since the range is 42, and the maximum value is 24, the minimum value can be found by subtracting 42 from the maximum value: 24 - 42 = -18.
Step 4: Determine the remaining 17 numbers
Since the mean is approximately 24, and the sum of all numbers in the set (except the maximum) is 480 - 24 = 456, the sum of the remaining 17 numbers should be 456. To make each number different from one another, we can construct a sequence where each number differs by 1. Starting from the minimum value (-18), we can add 1 repeatedly until the sum of all numbers reaches 456:
-18 + (-17) + (-16) + ... + (-3) + (-2) + (-1) + 0 + 1 + 2 + 3 + ... + 14 + 15 = 456.
Hence, the set of 20 numbers that satisfies the given conditions is:
[-18, -17, -16, ..., -3, -2, -1, 0, 1, 2, 3, ..., 14, 15, 24].
Note: The set of numbers provided in the answer satisfies all the given conditions, including the requirement of having no more than three numbers that are the same.
To find a set of numbers that satisfy these conditions, we can break down the process into steps and work through each condition one by one.
1. The median of a set of 20 numbers is 24:
To find a set of numbers with a median of 24, we need to consider that there are 10 numbers below 24 and 10 numbers above 24. For simplicity, let's start with 10 numbers below and 10 numbers above.
2. The range is 42:
The range is the difference between the largest and smallest numbers in the set. To have a range of 42, we need a large difference between the smallest and largest numbers. Since we want 10 numbers below 24, we can choose the lowest number to be 24 - 42 = -18.
3. The mean is 24 (to the nearest whole number):
The mean is the average of all the numbers in the set. Since there are 20 numbers in total, and we want the mean to be 24, the sum of all the numbers should be 20 * 24 = 480.
4. No more than three numbers are the same:
To ensure that no more than three numbers are the same, we can introduce some variability. We will randomly select three sets of 2 numbers to be the same, and the rest will be different.
Now, let's follow these steps to construct the set of numbers:
Step 1: 10 numbers below 24.
Start with -18, then add 2 to get -16, and so on, until we reach 14. This gives us the range of -18 to 14.
Step 2: 10 numbers above 24.
Start with 26, then add 2 to get 28, and so on, until we reach 46. This gives us the range of 26 to 46.
Step 3: Adjusting for the mean.
To find the sum of the numbers, we can sum up the numbers from each set separately and then adjust them to reach the desired mean.
For the set of 10 numbers below 24, the sum is:
-18 + -16 + ... + 12 + 14 = -20.
To adjust the sum to reach 480, we need to distribute the remaining sum of 480 - (-20) = 500 among the 10 numbers above 24. We can distribute this evenly, which would add 50 to each number. Therefore, the set of 10 numbers above 24 becomes:
26 + 50, 28 + 50, ..., 46 + 50.
Now, we have the numbers: -18, -16, ..., 14, 76, 78, ..., 96.
Step 4: Introduce variability.
To ensure no more than three numbers are the same, we'll randomly select three sets of 2 numbers from -16 to 14 and add them to our set. For example, we can select (-16, -14), (-10, -8), and (4, 6) to add some variability.
With these additional sets, our final set of 20 numbers that satisfy all the given conditions can be:
-18, -16, -14, -16, -10, -8, -8, -10, -2, 0, 2, 4, 4, 6, 6, 8, 10, 12, 14, 76, 78, ..., 96.