Business Statistics: The Ultimate Guide

The data below shows the time I take to walk to school on a weekday. 

Day	Time take (Minutes)
Monday	38
Tuesday	50
Wednesday	44
Thursday	42
Friday	49

Mean, 

and N = is an integer 

 =  44.6 

Therefore the mean equals to 44.8 minutes 

Median 

Refers to the middle value for data arranged in a sequence 

38, 42, 44, 49, 50 

Median is 44 

Range 

The range is the difference between the maximum and the minimum value 

Range = 

Range is 12 

 Standard Deviation, σ 

	X-  , where  = 44.8	2
38	-6.6	43.56
50	5.4	29.16
44	-0.6	0.36
42	-2.6	6.76
49	4.4	19.36
		∑ =99.2

Variance = 

Thus, standard deviation= 

Questions 

If you increase the highest original data value by 10 and decrease the lowest initial data value by 5, what measure changes; the mean, median, both, or neither? Why? 

The mean changes but the media remain the same. If any value is changed in a set of data, the mean also changes. 

Proof 

Let's add 5 and 10 to the lowest and highest values of our data in part A. The data will look as shown below; 

33, 60, 44, 42, 49 

The median is still 44 

Mean,  =  45.6 

The mean has changed by 1 (45.6 – 44.6) 

If you add 5 to each original data value, what measure changes; the standard deviation, range, both, or neither? Why? 

When five is added to each of the original data, there will be no changes to the standard deviation as well as the range. When a constant is added to the each of the original data, only the mean and the quartiles are affected. When calculating the standard deviation using the new mean the standard deviation will be the same. 

Proof 

	X-  , where  = 49.6	2
43	-6.6	43.56
55	5.4	29.16
49	-0.6	0.36
47	-2.6	6.76
54	4.4	19.36
		∑ =99.2

Variance = 

Standard deviation =  = 4.4542 

Range = 55-43=12 

Thus, the standard deviation and the range will not change when a constant is added to each set of the data. 

If you multiply each original data value by 5, what measure changes; the standard deviation, range, both, or neither? Why? 

Both will change. When each of the original data is multiplied by 5, both the range and the standard deviation will be multiplied by 5. 

Proof 

	X-  , where  = 223	2
190	-33	1089
250	27	729
220	-3	9
210	-13	169
245	22	484
		∑ =2480

Range = 250 – 190 = 60 this is 5 times the range of the original data (60 = 12x5) 

Variance = 

Standard deviation =  = 22.271 this is five times the standard deviation of the original data (4.4542x5) 

Week 3 

The sampling distribution of the mean is a theoretical distribution which is approached when the samples are increased relative to the frequency distribution. As the number of samples is increased the relative frequency distribution gets close too. The relative frequency distribution approaches sampling distribution as the number of samples approaches infinity. This section will carry out interactive simulation on sampling distributions. 

To run the virtual sampling distribution, “custom’ was selected and with the help of a mouse a distribution was traced. The distribution found is as shown below in the graph. The value of the mean, median, standard deviation, skew, as well as kurtosis is displayed alongside the graph. 

N was set to 25 (N = 25), and then the animate button was clicked. The graph obtained is as shown below. The value of the mean, median, standard deviation, skew, as well as kurtosis is displayed alongside the graph. 

Five was clicked to see the process repeated five times. The graph obtained is as shown below. The value of the mean, median, standard deviation, skew, as well as kurtosis is displayed alongside the figure. 

10, 000 was clicked to see the process repeated 10,000 times. The graph obtained is as shown below. As displayed in the diagram, the sampling distribution graph is approximately normal. The value of the mean, median, standard deviation, skew, as well as kurtosis is displayed alongside the chart. 

Week 4 

Percentage of Foreign-Born Residents in Philadelphia Compared with other Cities, 2016 

The data below shows a comparison in percentages of foreign-born residents in Philadelphia to other cities in the word in 2016. Foreign-born share ranks the towns. 

City	The foreign-born share of city population (%)
United States	13.5
Baltimore	8.1
Washington	13.3
Portland	13.7
Philadelphia	14.8
Denver	14.9
Minneapolis	15.4
Seattle	18.7
Boston	28.9
New York	37.5
San Jose	39.3

Average = 19.83 

Standard Deviation = 10.50 

Sample Size = 11 

Confidence Interval = 1.95 

Margin of Error, E = 6.20 

Upper Bound = 26.03 

Lower Bound = 13.62 

Max = 39.3 

Min = 8.1 

Range = 31.2 

4 stdve below = -22.16 

4 stdev above = 61.2 

N=11 

E = 6.2 

P = 0.95 

From table; C 95 = 1.960 

Critical Values = -1.960 and +1.960 

Z = 1.96 

P = 19.83 

The blue region shows the 95% confidence interval. 

References 

PEW. (2018). Philadephia’s Immigrants. [Online]. Available at: http://www.pewtrusts.org/en/research-and-analysis/reports/2018/06/07/philadelphias-immigrants . Accessed 3 rd Sep 2018.