Statistical Analysis: Substance Concentration, Potassium Levels, and Heights
Statistical Analysis of Data Sets
Substance Concentration in Packaging
The following data pertains to the content in milligrams per liter of a substance in forty packages:
250 215 185 235 220 255 230 165 210 180
175 225 185 220 160 285 225 260 185 205
235 200 230 245 180 205 195 175 180 190
195 240 170 195 250 220 190 215 270 205
Basic Interval for the Value 195
The elementary interval value 195 is (192.5, 197.5).
Grouping Values into Six Intervals
Grouping the values into six intervals, so that the apparent lower limit of the first interval is 150. Amplitude of each interval: (287.5 – 147.5) / 6 = 23.3. Rounded up to 25 mg / l. If the apparent lower limit of the first interval has to be 150, the actual limits are: (147.5, 172.5) (172.5, 197.5) (197.5, 222.5)
Table of Limits, Class Marks, and Frequencies
Table with actual and apparent limits, the class marks and the four types of frequencies:
| Limits (Real) | Limits (Apparent) | Class Marks | Ni | Fi | xini | |
|---|---|---|---|---|---|---|
| (147.5, 172.5) | 150 – 170 | 160 | 3 | 0.075 | 3 | 0.075 |
| (172.5, 197.5) | 175 – 195 | 185 | 13 | 0.325 | 16 | 0.400 |
| (197.5, 222.5) | 200 – 220 | 210 | 10 | 0.250 | 26 | 0.650 |
| (222.5, 247.5) | 225 – 245 | 235 | 8 | 0.200 | 34 | 0.850 |
| (247.5, 272.5) | 250 – 270 | 260 | 5 | 0.125 | 39 | 0.975 |
| (272.5, 297.5) | 275 – 295 | 285 | 1 | 0.025 | 40 | 1.000 |
| 40 | 1.000 |
Potassium Concentration in Packages
Measure the Potassium Concentration in thirty packages of a product and the result in milligrams per liter, were as follows:
570 630 650 570 590 640 600 620 690 650
670 620 630 700 620 590 640 580 720 590
660 610 690 590 600 640 680 730 650 660
Elementary Interval for the Value 620
The elementary interval value 620 is (615, 625)
Grouping Values into Seven Intervals
Groups values into seven intervals, so that the apparent lower limit of the first interval is 550 and make a table of actual and apparent limits, the class marks and the four types of frequencies. Amplitude of each interval: (735 – 545) / 7 = 27.1. Rounded up to 30 mgr / l.
Table of Limits, Class Marks, and Frequencies
| Limits (Real) | Limits (Apparent) | Class Marks | Ni | Fi | xini | |
|---|---|---|---|---|---|---|
| (545, 575) | 550 – 570 | 560 | 2 | 0.067 | 2 | 0.067 |
| (575, 605) | 580 – 600 | 590 | 7 | 0.233 | 9 | 0.300 |
| (605, 635) | 610 – 630 | 620 | 6 | 0.200 | 15 | 0.500 |
| (635, 665) | 640 – 660 | 650 | 8 | 0.267 | 23 | 0.767 |
| (665, 695) | 670 – 690 | 680 | 4 | 0.133 | 27 | 0.900 |
| (695, 725) | 700 – 720 | 710 | 2 | 0.067 | 29 | 0.967 |
| (725, 755) | 730 – 750 | 740 | 1 | 0.033 | 30 | 1.000 |
| 30 | 1.000 |
Median and Quartiles
Calculate the median and the quartiles with pooled data, and draw a box plot and histogram aligned.
Q1 = 575 + (7.5 – 2) / 7 * 30 = 598.6
Md = 635 + (15 – 15) / 8 * 30 = 635.0
Q3 = 635 + (22.5 – 15) / 8 * 30 = 663.1
Height Ranges in Groups
Heights in groups the following ranges, with the criteria you consider appropriate:
1.53 1.62 1.60 1.68 1.71 1.63 1.82 1.66 1.61 1.69
1.67 1.45 1.75 1.73 1.66 1.79 1.72 1.74 1.56 1.71
1.77 1.68 1.64 1.69 1.72 1.56 1.67 1.64 1.70 1.61
1.57 1.65 1.65 1.63 1.62 1.72 1.70 1.64 1.46 1.70
Interval Calculation
Number of intervals: N = 40 = 6.32. Rounded to six intervals. Amplitude of each interval: (1.825 – 1.445) / 6 = 0.063. Rounded up to 0.07. Starting with 1.45, the table would read:
Table of Real Limits, Class Marks, and Frequencies
| Limits (Real) | Class Marks | Limits (Apparent) | Frequency (Absolute) |
|---|---|---|---|
| (1.445, 1.515) | 1.48 | 1.45-1.51 | 2 |
| (1.515, 1.585) | 1.55 | 1.52-1.58 | 4 |
| (1.585, 1.655) | 1.62 | 1.59 to 1.65 | 12 |
| (1.655, 1.725) | 1.69 | 1.66 to 1.72 | 16 |
| (1.725, 1.795) | 1.76 | 1.73-1.77 | 5 |
| (1.795, 1.865) | 1.83 | 1.80 to 1.86 | 1 |