Question

I need help finding the five-number summary of the given numbers

Answer 1

We are given the following numbers

3 8 8 4 2 9 8 7 9 5 9 3 2 5 2 9 8 7 7 9

to calculate the five number summary, it means that we should calculate the following numbers:

- minimum

- percentile 25

- median (or percentile 50)

- quantile 75

- maximum

So, we should first sort this numbers from least to greatest, so it is easier to identify this five numbers. Once sorted we get the following numbers:

2 2 2 3 3 4 5 5 7 7 7 8 8 8 8 9 9 9 9 9

From this arrangement, we can see that the least number (or minimum) is 2 and the greatest number (or maximum) is 9.

Now we need to identify the percentiles 25, 50 and 75.

To calculate this numbers, we must first count the number of data points we have. Once we do so, we get that we have 20 elements.

Now, the term percentile 50 means that we are finding a point such that 50% of the data (the half) is on both sides of the number.

So, since we have 20 data points, we must find a number that splits the data in a way such that 10 points are less than or equal to the number and 10 points are greater than or equal to the number.

So, first, we count 10 numbers from left to right. The 10th number in the sorted sequence is 7, and the 11th number is also 7. So, the number that we are looking for is the average of this two numbers. So we get

`(7+7)/(2)=7`

So the percentile 50 is 7.

Now, we need to find the percentile 25 and percentile 75.

The percentile 25 is the number such that 25% of the data is less than or equal to this number.

Note that since we have 20 data points. The 25% of 20 is 5. That is, we should find a number such data 5 data points is on the left of the number and 15 numbers are on the right of the number. If we count 5 numbers, we get the number 3. The 6th number is 4. So, the number we are looking for should be inbetween 3 and 4. ONe way to find this number is to take a weighted average of this numbers. Since we are looking for the percentile 25, we will give a weight of 0.25 to 3 and 0.75 to 4. Then the percentile 25 would be

`0.25\cdot3+0.75\cdot4\text{ =3.75}`

In the same manner, to find the percentile 75, we first identify the 15th and 16th numbers. In this case, the 15th number is 8 and the 16th number is 9. Since we want the percentile 75, we give a weight of 0.75 to 8 and a weight of 0.25 to 9. Then we get

`0.75\cdot8+0.25\cdot9\text{ = 8.25}`

Then, the percentile 75 is 8.25.

Then, the five summary of the data is

- minimum : 2

- percentile 25 : 3.75

- median (or percentile 50): 7

- quantile 75: 8.25

- maximum: 9