Minitab
• How to open a data file?
• How is the data stored?
• Generate Histogram
Numerical Measures
• Center (where is middle?)
• Variation (how much does it vary?)
Measures of Center
n
• (Sample) Mean
x
x1  x2  ...  xn
x

n
n
i 1
• (Sample) Median
– the number in the middle
i
Example
• Birth weight of five babies born in a
hospital: 9.2, 6.4, 10.5, 8.1, 7.8
• Mean birth weight =
(9.2+6.4+10.5+8.1+7.8)/5= 8.4
• Median
6.4 7.8 8.1 9.2 10.5
Another Example
• Survival Time: 3 15 46 64 126 623
• Mean survival time =
(3+15+46+64+126+623)/6 = 146.2 days
• Median
– (46+64)/2 = 55 days
• Median less affected by extreme
observations (outliers)
• Median is more sensible measure for
extreme asymmetrical data
Sample 100p-th Percentile
1. Order the data from the smallest to largest
2. Determine np
3. If np is not an integer, round it up, say k,
and find the kth ordered value.
4. If np is an integer, say k, find the average
of the kth and the (k+1)th ordered value.
Sample Quartiles
• First (Lower) Quartile Q1= 25th percentile
• Second Quartile (Median) Q2= 50th percentile
• Third (Upper) Quartile Q3= 75th percentile
Traffic Noise Level in Decibels
52.0 54.4 54.5 55.7 55.8 55.9 55.9 56.2 56.4
56.4 56.7 56.8 57.2 57.6 58.9 59.4 59.4 59.5
59.8 60.0 60.2 60.3 60.5 60.6 60.8 61.0 61.4
61.7 61.8 62.0 62.1 62.6 62.7 63.1 63.6 63.8
64.0 64.6 64.8 64.9 65.7 66.2 66.8 67.0 67.1
67.9 68.2 68.9 69.4 77.1
Measures of Variation
Two data set:
12345
23334
Observation
x
Deviation
xx
Squared deviation
1
-2
4
2
-1
1
3
0
0
4
1
1
5
2
4
Total = 15
Total = 0
Total = 10
Variance and Standard Deviation
• Sample Variance
s2 = (sum of squared deviations) /(n-1)
n
s 
2
 ( x  x)
i 1
2
i
n 1
• Sample Standard Deviation
s = Square Root of Variance
• Sample Variance
s2 = 10/(5-1)=2.5
• Sample Standard Deviation
s  2.5  1.58
68-95-99.7 rule
If the distribution is bell sharp, then approximately
– 68%
of the data lie with
xs
– 95%
of the data lie with
x  2s
– 99.7% of the data lie with
x  3s
Boxplot
• Five Number Summary
–
–
–
–
–
Minimum
Q1
Median
Q3
Maximum
• Inter-Quartile Range: Q3-Q1