University of Technology Sydney
School of Mathematical & Physical Sciences
Mathematics 2 (33230) Lab 1 (Statistics) Solutions
Question 1
a)
Discuss the apparent effectiveness of the three different methods in terms of the
concepts of accuracy and precision.
It appears that both Methods 1 and 3 are accurate, that is, the methods are likely to
obtain a measurement of 80. Method 2 is not accurate in this way. Methods 2 and 3
appear to be precise, that is, have a relatively low IQR. Measurements from Method 1
do not appear to be as accurate, as the range and IQR are larger.
b)
Explain how the summary statistics obtained here support the conclusions that you
made in 1(a).
Method 1
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Method 2
80.2
1.280624847
79
78
4.049691346
16.4
-0.662764511
0.370900494
13
74
87
802
10
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Method 3
84.2
0.553775
84.5
85
1.75119
3.066667
-1.00965
-0.06828
5
82
87
842
10
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
81.5
0.718795
82
82
2.27303
5.166667
-0.8839
-0.14192
7
78
85
815
10
We can see that the means for both Methods 1 and 3 are close to 80, so the measurements are
accurate. The mean for Method 2 is further away from 80. The standard deviations for Methods
2 and 3 are smaller than the standard deviation from Method 1, so the measurements for
Methods 2 and 3 are more precise. These are the same conclusions as those obtained using the
multiple boxplots.
33230 LAB 1 SOLUTIONS
1
Question 2
a)
Are there any obvious time trends or cycles in the data? What practical engineering
reason is there for looking at such trends?
No obvious trends or cycles in the data. In a production situation, we are usually
looking for trends, which suggest that the production process is deteriorating over time.
We may also be looking for cycles, which affect the consistency of measurements.
b)
Describe the shape of the histogram in terms of its centre, spread and skewness.
This distribution seems to be centred at about 150, and ranges from 30 to about 300,
with a single potential outlier at about 500. The distribution of the measurements is
skewed to the right.
c)
What are the median, first and third quartiles for the aluminium readings?
Quartile
0th aka Minimum or 0%
1st aka 25%
2nd aka Median or 50%
3rd aka 75%
4th aka Maximum or 100%
Value
30
87.75
119
197.5
511
Command
=QUARTILE(B2:B27,0)
=QUARTILE(B2:B27,1)
=QUARTILE(B2:B27,2)
=QUARTILE(B2:B27,3)
=QUARTILE(B2:B27,4)
The median is 119, Q1 = 87.75, and Q3 = 197.5.
33230 LAB 1 SOLUTIONS
2
d)
Are the transformed data more symmetric than the untransformed data?
After we take logs, the data appears more symmetric.
e)
Is the mean of the transformed values equal to the natural logarithm of the mean of the
original data?
Aluminium
Mean
Standard Error
Median
Mode
Standard
Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Ln(Al)
142.6538
19.25944
119
119
98.20425
9644.075
7.283683
2.284891
481
30
511
3709
26
Mean
Standard Error
Median
Mode
Standard
Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
4.772864
0.123836
4.779123
4.779123
0.631443
0.39872
0.887851
-0.19893
2.835172
3.401197
6.23637
124.0945
26
As ln(142:7) = 4:96, the mean of the transformed data is not equal to the log of the
mean of the original data.
33230 LAB 1 SOLUTIONS
3
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