Question

Given the data set A = {9, 5, 16, 4, 32, 8, 12, 9, 11, 15, 5, 9, 18, 11}, which is the data of an entire population of subjects: Calculate the arithmetic mean (5 pts) Find the median (5 pts) Find the mode (5 pts) Calculate the range (5 pts) Calculate the interquartile range (5 pts) Calculate the mean deviation (5 pts) Calculate the variance (5 pts) Calculate the standard deviation (5 pts)

Answer 1

Answer:

1. Arithmetic Mean:  11.71

2. Median:  10

3. Mode:  9

4. Range:  28

5. Interquartile Range (IQR):  7

6. Mean Deviation:  4.92

7. Variance:  47.63

8. Standard Deviation:  6.90

Step-by-step explanation:

Lets first write the set of numbers from smallest to largest:

A = {4, 5, 5, 8, 9, 9, 9, 11, 11, 12, 15, 16, 18, 32}

1.

Arithmetic Mean:

This is the sum of all the number divided by the number of numbers.

So we can write:

Arithmetic Mean = $\frac{9+5+16+4+32+8+12+9+11+15+5+9+18+11}{14}=\frac{164}{14}=11.71$

Hence,

Arithmetic Mean = 11.71

2.

Median:

In a set of numbers, the median is the "middle" value of the numbers when arranged from smallest to largest.

If there is n numbers and there are odd number of numbers, the median would be:

(n+1)/2 th number

If there is n numbers and there are even number of numbers, the median would be:

Average of n/2th number and (n/2)+1 th number

Since here are even number of numbers (n= 14), so we take the average of:

7th and 8th number

Looking at the list, we see

7th number = 9

8th number = 11

So,

Median = (9 + 11)/2 = 10

3.

Mode is the the number that occurs the "most" number of times.

If there is no number that occurs more than others, we can say the list has NO MODE.

If there are 2 numbers that occur the most and same number of times, we say that it is BIMODAL.

If there are more, we can say it is MULTIMODAL.

Now, looking at the list, we see "9" occurs the most number of times, THREE TIME.

So,

Mode = 9

4.

THe range is the difference in maximum value and the minimum value, or the highest number MINUS the lowest number.

Looking at the list:

Max = 32

Min = 4

So,

Range = Max - Min = 32 - 4 = 28

Range = 28

5.

Interquartile Range (IQR):

This is the difference in 3rd quartile and the 1st quartile. We can say:

$IQR=Q_3 - Q_1$

Q_1 (First Quartile): For even number of number (here, n = 14), the first quartile would be

n/4 = 14/4 = 3.5

So average of 3rd and 4th number. That would be:

(5+8)/2=13/2=6.5

Q_3 (Third Quartile): For even number of numbers (here, n = 14), the third quartile would be:

(3/4) * n = (3/4) * 14 = 10.5

So average of 10th and 11th number. That would be:

(12+15)/2 = 27/2 = 13.5

IQR = Q_3 - Q_1 = 13.5 - 6.5 = 7

Inter Quartile Range = 7

6.

Mean Deviation:

This is the deviation of each value from the mean. We first subtract the mean from each of the values and then take the absolute value of those numbers. Then we take those deviations' mean.

Mean was 11.71

Deviations:

4 - 11.71 = -7.71

5 - 11.71 = -6.71

5 - 11.71 = -6.71

9 - 11.71 = -2.71

9 - 11.71 = -2.71

9 - 11.71 = -2.71

11 - 11.71 = -0.71

11 - 11.71 = -0.71

12 - 11.71 = 0.29

15 - 11.71 = 3.29

16 - 11.71 = 4.29

18 - 11.71 = 6.29

32 - 11.71 = 20.29

We take the mean of the absolute value of the deviations:

Mean Deviation = (7.71+6.71+6.71+2.71+2.71+2.71+0.71+0.71+0.29+3.29+4.29+6.29+20.29)/14 = 4.92

Mean Deviation = 4.92

7. Variance:

This is the measurement of spread between the numbers is a data set.

To get variance, we take the Mean Deviations we got in previous part, SQUARE them, Take the Sum, and then divide that sum by n (n = 14).

So,

$Variance = \frac{7.71^2+6.71^2+6.71^2+2.71^2+2.71^2+2.71^2+0.71^2+0.71^2+0.29^2+3.29^2+4.29^2+6.29^2+20.29^2}{14}=47.63$

Variance = 47.63

8.

Standard Deviation:

THis is a qunatity that tells us "how much" the numbers in a set of values DIFFER from the mean value. Also a measure of dispersion and variability.

The standard deviation is found by taking the Square Root of the variance.

In the previous part, we found:

Variance = 47.63

Hence, Standard Deviation would be:

$\sqrt{47.63 }= 6.90$