Question

The data below represent the cost in dollars that each of 40 cafeteria customers paid for their salad, as observed by statistics students in Fall 2018.

2.45	2.89	3.11	3.34	3.38	3.80	3.83	3.88	3.93	4.06
4.09	4.19	4.23	4.29	4.33	4.38	4.75	4.96	5.00	5.08
5.20	5.25	5.28	5.45	5.65	5.70	5.72	5.78	5.81	5.93
5.98	6.15	6.28	6.40	6.56	6.59	6.79	6.90	7.08	7.24

• Compute the summary stats: mean, median, standard deviation, IQR, & range.
  • What can you say about the general shape of the distribution of salad costs? How do you know that?
  • Does this data set appear to contain any outliers? Explain.
• Omit the highest and lowest costs, and recompute the summary stats.
  • Why does the median remain unchanged?
  • Why do all the measures of variability decrease?
  • Why does the mean increase slightly?

Answer 1

The data is enterd into excel from column A1 to A40.

Mean	=AVERAGE(A1:A40)	5.04375
Median(Q2)	=MEDIAN(A1:A40)	5.14
s.d.	=STDEV(A1:A40)	1.241889973
Q1	=PERCENTILE(A1:A40,0.25)	4.0825
Q3	=PERCENTILE(A1:A40,0.75)	5.9425
IQR	=Q3-Q1	1.86
Range	=MAX(A1:A40)-MIN(A1:A40)	4.75
Q3-Q2	=Q3-Q2	0.8025
Q2-Q1	=Q2-Q1	1.0575
Inner Fence	=Q1-1.5*IQR	1.2925
Outer Fence	=Q3+1.5*IQR	8.7325

The shape of the distribution of the salad costs is approximately symmetric since Q3-Q2Q2-Q1.

Since the max value = 7.24 and min = 2.49, so there are no points outside the inner or outer fence. Hence there are no outliers.

After omitting the Highest and lowest costs the summary of the data is:

Mean	=AVERAGE(A1:A38)	5.053157895
Median(Q2)	=MEDIAN(A1:A38)	5.14
s.d.	=STDEV(A1:A38)	1.148451614
Q1	=PERCENTILE(A1:A38,0.25)	4.115
Q3	=PERCENTILE(A1:A38,0.75)	5.9
IQR	=Q3-Q1	1.785
Range	=MAX(A1:A38)-MIN(A1:A38)	2.85

Since median does not depends on the maximum and minimum values it only depend on the number of observation. So when they are deleted though the number of observation decreases but still it is even and hence the median remains same.

measures of variability depends on the values of the dataset. So when the max and the min values are deleted naturally the mean will be clustered where most of the data lies, so decreasing the variability.