PROBLEM 2: Average wait in minutes for the train QM3; n=12 15 4 5 9 6 12 17 10 8 13 8 16 example: of how it should be answer: PROBLEM: #colds / year n=18 subjects 10; 8; 5; 2; 3; 3; 3; 6; 4; 2; 4; 5; 4; 4; 1; 0; 3; 5; For computations done by hand, data must be ordered. 0 1 2 2 3 3 3 3 4 Median 4 4 4 5 5 5 6 8 10 Q1 Q3 Sample mean = =72/18 = 4 colds Sample median = 4 colds (midpoint of data) Sample mode = 3 and 4 (the value that occurs most often; this is a bimodal data set) First Quartile (approximation) = 0.25 * N = 0.25 * 18 = 4.5; If the value is not an integer, we can round it up to the nearest integer 5 à value is 3 Third Quartile (approximation) = 0.75 * N = 0.75 * 18 = 13.5 à 14 à value is 5 Range = max – min =10 – 0 = 10 IQR = Q3 – Q1 = 5 – 3 = 2 Variance = s2 = = 96/17 = 5.64 colds squared Standard deviation = s = = 2.37 colds Coefficient of variation = CV = x 100% =2.37/4 *100% = 59.25% To convert to a Z-score: Zi = (Xi - mean)/s The 0 becomes (0 – 4) / 2.37 = -1.68; the 1 becomes (1 – 4) /2.37 = -1.27; the 2 becomes (2- 4)/2.37 = .84; and the 4 becomes a 0; etc. All values below the mean have negative Z scores and all values above the mean have positive Z scores. This is the output from MS Excel using the descriptive tool. For your homework solution an output as the one below (for the excel solution) will be sufficient. Column1 Mean 4 Standard Error 0.560112 Median 4 Mode 3 Standard Deviation 2.376354 Sample Variance 5.647059 Kurtosis 1.497461 Skewness 0.887652 Range 10 Minimum 0 Maximum 10 Sum 72 Count 18 Quartiles are approximations. A more refined definition is the following: For Lower Quartile (25%): Sort all observations in ascending order Compute the position L1 = 0.25 * N, where N is the total number of observations. If L1 is already a whole number (integer), the lower quartile is midway (average) between the L1-th value and the next one. If L1 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the lower quartile. For Upper Quartile(75%): Sort all observations in ascending order Compute the position L3 = 0.75 * N, where N is the total number of observations. If L3 is a whole number, the upper quartile is midway (average) between the L3-th value and the next one. If L3 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the upper quartile. Please follow the above definitions in calculating the quartiles. There are two distinct situations: set size equal to a power of 4 (L is already a whole number – integer), set size not a power of 4.
PROBLEM 2: Average wait in minutes for the train QM3; n=12
15 4 5 9 6 12 17 10 8 13 8 16
example: of how it should be answer:
PROBLEM: #colds / year n=18 subjects
10; 8; 5; 2; 3; 3; 3; 6; 4; 2; 4; 5; 4; 4; 1; 0; 3; 5;
For computations done by hand, data must be ordered.
0 1 2 2 3 3 3 3 4
Q1 Q3
Sample mean = =72/18 = 4 colds
Sample median = 4 colds (midpoint of data)
Sample
First Quartile (approximation) =
0.25 * N = 0.25 * 18 = 4.5; If the value is not an integer, we can round it up to the nearest integer 5 à value is 3
Third Quartile (approximation) = 0.75 * N = 0.75 * 18 = 13.5 à 14 à value is 5
IQR = Q3 – Q1 = 5 – 3 = 2
Variance = s2 = = 96/17 = 5.64 colds squared
Standard deviation = s = = 2.37 colds
Coefficient of variation = CV = x 100% =2.37/4 *100% = 59.25%
To convert to a Z-score: Zi = (Xi - mean)/s
The 0 becomes (0 – 4) / 2.37 = -1.68;
the 1 becomes (1 – 4) /2.37 = -1.27;
the 2 becomes (2- 4)/2.37 = .84;
and the 4 becomes a 0; etc.
All values below the mean have negative Z scores and all values above the mean have positive Z scores.
This is the output from MS Excel using the descriptive tool. For your homework solution an output as the one below (for the excel solution) will be sufficient.
|
Column1 |
|
|
|
|
|
Mean |
4 |
|
Standard Error |
0.560112 |
|
Median |
4 |
|
Mode |
3 |
|
Standard Deviation |
2.376354 |
|
Sample Variance |
5.647059 |
|
Kurtosis |
1.497461 |
|
Skewness |
0.887652 |
|
Range |
10 |
|
Minimum |
0 |
|
Maximum |
10 |
|
Sum |
72 |
|
Count |
18 |
|
For Lower Quartile (25%):
|
For Upper Quartile(75%):
|
Please follow the above definitions in calculating the quartiles. There are two distinct situations: set size equal to a power of 4 (L is already a whole number – integer), set size not a power of 4.
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