Basic Statistics Terminology
Overview
Again I have to apologise beforehand to any statistics practitioners. In a previous article we looked at basic statistics terminology and addressed the expected value or mean of a set of determined values. In this article, we expand this to the variance, median, mode, and the very useful standard deviation. I hope, this article will offer people a little more grasp with the following terminologies.
Median:
In the recent article we checked out the 'mean' or 'expected value' of a set of values which we might measure. These were:
Values: 3, 5, 5, 6, 7, 9, 10, 11, 12, 12, 15, 16
Ordinarily, determined values would have a random order but the above values have been ordered from low to high.
The median value is the value at the center of the sequence that has 50 % of its values higher and 50 % of the values lower.
In the above occurrence there is no 'middle' value because we have 12 values. So, we select the 2 central values of 10 and 9 and average them to get 9.5. This is then the median value.
If we had the values: 3, 5, 5, 6, 7, 9, 9, 10, 11, 12, 12, 15, 16, the midpoint value (of a total number of 13) would have been 9 as the median value.
When values change a great deal the median can be beneficial as a tool that evens out the data. It can serve to follow trends in the data by way of noting the median values. Data values can at that point be deemed a difference from the median and can present a notion whether it is shifting beyond this trend.
If the median value is exactly the same as the mean or expected value at that point there is an even spread of values. Whenever the median is greater or less than the expected value or mean, then the distribution of the values will be slanted either towards the left or right.
Mode:
When it comes to basic statistics terminology this is basic. If we once more evaluate the values above measured and adjust one of them from 15 to 12 we have:
Values: 3, 5, 5, 6, 7, 9, 10, 11, 12, 12, 12, 16
The mode is the value which occurs the most times, in the above situation it will be 12, which appears in 3 places.
Also there can be 2 modes or more in a set of values.
Variance:
If you recall, the mean was additionally referred to as the 'expected value'. Every measured data value will differ from this expected value or mean by a precise amount. The variance presents a notion of how 'stretched out the values are' when equated to the mean or expected value.
The total variance amounts to the average of the sum of the individual variances.
The variance is worked out as the square of the deviation between it and the expected value or mean. For example:
Data values: 3, 5, 5, 6, 7, 9, 10, 11, 12, 12, 15, 16
Mean or expected value: (3 + 5 + 5 + 6 + 7 + 9 + 10 + 11 + 12 + 12 + 15 + 16)/12 = 111/12 = 9.25
If we look at 6 (the 4th value) the variance will be:
Variance = (6-9.25) x (6-9.25) = (-3.25) x (-3.25) = 10.56
We can calculate this for each of the data values, total these up then divide by the total number of values, 12 to get the overall variance.
We could employ this concept for a simplified project task delay in the previous article:
Delay..........Probability..........Contribution
6......................0.3...................6 x 0.3 = 1.8
16....................0.5..................16 x 0.5 = 8.0
20....................0.2...................20 x 0.2 = 4.0
The expected value = 1.8 + 8.0 + 4.0 = 13.8 working weeks
The overall variance will be the total of the individual variances divided by 3, the number of values.
Total variance = [(6-- 13.8) x (6-- 13.8) x 0.3 + (16-- 13.8) x (16-- 13.8) x 0.5 + (20-- 13.8) x (20-- 13.8) x 0.2]/3
= [(-7.8) x (-7.8) x 0.3 + (2.2) x (2.2) x 0.5 + (6.2) x (6.2) x 0.2]/3
= [(60.84 x 0.3) + (4.84 x 0.5) + (38.44 x 0.2)]/3
= (18.25 + 2.42 +7.69)/3
= 28.36/3
= 9.45
This gives a notion of the distribution of values in relation to the 'expected value'.
Note that in this illustration the 'values' were defined by an expert's assessment hinged upon assumptions, therefore these are not 'measured values'. For measured values we would have had to, in fact, perform the activity three times in just the exact same way and established that, on those three particular circumstances, the hold ups were 6, 16 and 20 working weeks. This could not occur in the real world.
Standard deviation:
This is the square root of the variance. For the above we have:
Standard deviation = √9.45 = 3.07
This is a really useful value.
When we determine data values there will be a 68 percent chance that all of the values will occur inside 1 standard deviation of the mean or expected value.
For the example above:
Mean or expected value = 13.8
Variance = 9.45
Standard deviation = 3.07
68 % of data values will fall within (13.8-- 3.07) and (13.8 + 3.07) = 10.73 to 17.5
In an identical way 95 percent of the data values will occur inside 2 standard deviations and 99.7 percent will fall within 3 standard deviations. So, we would have:
2 standard deviations = 6.14
3 standard deviations = 9.21
95 % of values will occur inside (13.8-- 6.14) and (13.8 + 6.14) = 7.66 to 19.94
99.7 percent of values will land inside (13.8-- 9.21) and (13.8 + 9.21) = 4.59 to 23.01
Ideally, this article has given a modest insight into a few statistical expressions.
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