ECET 3710: C# Statistical Analysis – Part 9
Average (mean):
̅
∑
=μ
Example: Given x = {67,98,22,100,62,84,73}
̅
Median: middle value of an ordered (smallest to largest) data set. Unlike the mean it can be
useful when the data set contains outliers.
Example: Given x = {22,62,67,73,84,98,100}
If x contains an odd number of elements then choose the middle number:
M = 73
Else if, x contains an even number of elements, then average the two middle numbers.
Standard Deviation: typical deviation from the mean of a data set.
√∑
̅
Example: Given x = {22,62,67,73,84,98,100}, ̅
(22 – 72.3)2 + (62 – 72.3)2 + (67 – 72.3)2 + (73 – 72.3)2 + (84 – 72.3)2 + (98 – 72.3)2 + (100 – 72.3)2
= 4229.43
4229.43/6 = 704.905
σ=√
= 26.55
Class example: Given x = {48,57,56,64,63,52,47,50}, calculate σ
Question: Based on the two standard deviations calculated (instructor and class examples)
explain what σ tells us about the numbers in the data set.
Question 2: Given a sampled data set what can be deduced about the data for large values of
σ? What can be deduced for small values of σ?
Question 3: An embedded system measure the humidity in an equatorial rain forest with a
sampling rate of one hour over a period of several days. What might you expect σ to be?
Empirical Rule (68-95-99.7): 68% of the data set lies within 1σ (plus or minus) of the mean; 95%
of the data set lies within 2σ (plus or minus) of the mean; 99.7% of the data set lies within 3σ
(plus or minus) of the mean.
Range: difference between the highest and lowest values in a data set.
Example: Given x = {22,62,67,73,84,98,100}
Range = 100 – 22 = 78
Percentiles: a measure of relative standing within the overall data set. The kth percentile, for
instance, refers to located in a sub-sample containing k percent of the data set. The rest of the
data is contained within [100 – k] percent of the data set.
Example: Given x = {22,62,67,73,84,98,100}, Find the 30th percentile
0.30 x n, where n = number of elements in the data set; 0.3 x 7 = 2.1 round this number up to
3 counting from the left arrive at the 3rd number, 67  therefore, 67 is the 30th percentile.
Note: If the product of the desired percentile and n is a whole number, then count from the left
as before, but average the number at that position and the one following it. For example, if we
were seeking the 30th percentile in a data set of 10 numbers, then we would multiply 0.3 x 10 =
3. From the left we would average the third and fourth number to calculate the 30th percentile.
Question: Given x = {24,27,3,35,36,40,44,47,52,64,67,71,79,80,82,88,92,93,94}, find all
numbers between and including the 30th and 40th percentile.
Interquartile Range (IQR): range of numbers within the inner 50% (75th percentile minus the
25th percentile) of a data set.
Question: What is the IQR for the preceding data set?
Normalization: Several normalization schemes exist. However, one of the most common is to
divide all values in the data set by the highest value, thus scaling all values to a range of 0 to 1.
Question: What is the normalized version of the preceding data set?
Question 2: Why might you normalize a data set?