Question

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed. 42 36 48 51 39 39 42 36 48 33 39 42 45 50

Answer 1

here are the descriptive statistics for the given sample :

The sample size is n = 14. The provided sample data along with the data required to compute the sample mean X and sample variance sº are shown in the table below: annual salary annual salary2 42 1764 36 1296 48 2304 51 2601 39 1521 39 1521 42 1764 36 1296 48 2304 33 1089 39 1521 42 1764 45 2025 50 2500 Sum 590 25270

The sample mean X is computed as follows: X=- n X; 590 14 = 42.143 i=1 Also, the sample variance s2 is 1 1 (Ex-+ (2x) 25270 5902 14 = 31.209 n 1 14-1 Therefore, the sample standard deviation s is S = V 32= 31.209 = 5.586 Now, in order to compute the median and the quartiles, the data needs to be put into ascending order, as shown in the table below Position annual salary (Asc. Order) 1 33 2 36 3 36 4 39 5 39 6 39 7 42 8 42 9 42 10 45 11 48 12 48 13 50 14 51

Since the sample size n = 14 is even, we have that (n+1)/2 = (14+1)/2 = 7.5 is not an integer value, the median is computing by taking the average of the values at the positions 7th and 8th, as shown below median 42 +42 2 = 42 Quartiles The quartiles are computed using the table with the data in ascending order For Qı we have to compute the following position: 25 25 pos(Q1) = (n +1) = (14+1) = 3.75 100 100 = Since 3.75 is not an integer number, Q1 is computed by interpolating between the values located in the 3th and 4th positions, as shown in the formula below Q1 = 36+ (3.75 – 3) < (39 – 36) = 38.25 For Q3 we have to compute the following position: 75 75 pos(Q3) = (n +1) (14+1) 100 100 = 11.25 Since 11.25 is not an integer number, Q3 is computed by interpolating between the values located in the 11th and 12th positions, as shown in the formula below Q3 = 48 +(11.25 – 11) × (48 – 48) = 48 Also, from the information above we get that min = 33, max = 51, which means that the range is Range = max - min = 51 – 33 = 18 Finally, the 5-Number summary is min = 33, Q1 = 38.25, median = 42,Q3 = 48, max = 51

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