Measure of Variations

example:

If we have two data sets:

A: 1, 4, 5, 6, 9

B: 5, 5, 5, 5, 5

The mean = $\bar{x} = \frac{\sum x}{n}$ → $\bar{x} _A =\bar{x} _B=$ 5

<aside>

Here, we have two different data sets but with an equal mean, which indicates that using the mean solely is insufficient to evaluate the data and reach a conclusion.

</aside>

1. Range

The range of a data set is the size of the narrowest interval, which contains all the data.

For the same previous example:

A: 1, 4, 5, 6, 9

B: 5, 5, 5, 5, 5

The mean = $\bar{x} = \frac{\sum x} { n}$ → $\bar{x} _A =\bar{x} _B=$ 5

The range = Max - Min → $Range_A =$ 8 and $Range_B=$ 0

<aside>

Both Data sets have equal mean but differing ranges

</aside>

However, this is not always the case. For the following example:

A: 2, 4, 6, 8

B: 2, 2, 8, 8

The mean = $\bar{x} = \frac{\sum x}{ n}$ → $\bar{x} _A =\bar{x} _B=$ 5

The range = Max - Min → $Range_A = Range_B =$ 6

<aside>

Both Data sets have equal mean and range

</aside>

2. Variance

The variance measures the dispersion of data with respect to the mean.

       **In the second figure, the data is more scattered about the mean than in the first figure**

• The equation of variance:

  • Population: $\sigma^2 = \frac{ \sum(X-\mu)^2}{N}$
  • Sample: $S^2 = \frac{\sum (x-\bar{x})^2}{n-1}$

• The Equation of Standard deviation:

  The standard deviation is the square root of the variance for both population and sample

  • $\sigma=\sqrt{\sigma^2} / S=\sqrt{S}$

<aside>

The calculator can calculate both variance and standard deviation. Please see this video to learn how to use it.

</aside>

https://www.youtube.com/watch?v=AD_e7qW_Qq0

Coefficient of Variation

The Coefficient of Variation (CV) is a statistical measure of the relative variability of data, often used to compare the degree of variation between datasets with different means.

$$

CV = \frac{\sigma}{\mu}*100 / \frac{s}{\bar{x}}*100

$$

Example:

Imagine you have two performance metrics for a web service:

• API Response Times (in milliseconds): 100, 95, 105, 110, 90
• Memory Usage (in MB): 500, 490, 515, 505, 510

You want to compare how variable these metrics are relative to their mean values.

We will use the calculator as shown in the video

1. API response time: $\bar{x} = 100 ms$ and $S=7.91ms$
2. Memory Usage: $\bar{x} = 504$ MB and $S=9.61$ MB

Then calculate the coefficient of variation:

1. $CV_1=\frac{7.91}{100}*100 = 7.91\%$
2. $CV_{Memory}=\frac{9.61}{504}*100 = 1.91\%$

<aside>

• Although response times and memory usage have different means and units, the coefficient of variation indicates that response times are more variable relative to their mean than memory usage is.
• The variation in memory usage (9.61 MB standard deviation) is large in absolute terms, but relative to a mean of 504 MB, it’s smaller (about 1.91%) compared to response times of 7.91% CV. </aside>

Chebyshev’s Theorem

It is an estimation of the minimum proportion of observations that will fall within a specified number of standard deviations regardless of the shape of the distribution.

The Empirical (Normal) Rule

It estimates the minimum proportion of observations that will fall within a specified number of standard deviations of normally distributed data.