Summary Statistics in Data Analysis

Summary statistics  are numbers that summarize properties of the data. i.e - Mean, Spread, tendency etc. We will see each one by one.

Let's take a input dataset -

Input: 45, 67, 23, 12, 9, 43, 12, 17, 91
Sorted: 9, 12, 12, 17, 23, 43, 45, 67, 91

Frequency: The frequency of an attribute value is the percentage of time the value occurs in the data set.
In our dataset, Frequency of 12 is 2.


Mode: The mode of a an attribute is the most frequent attribute value

Mode for our dataset is 2 as 12 is the most frequent item which occurs 2 time

Things to remember:
i- There is no mode if all the values are same
ii - Same is applicable if all values occurrence is 1 

Usually, Mode and Frequency are used for categorical data

Percentiles: This used for continuous data.
Given an ordinal or continuous attribute x and a number p between 0 and 100, the pth percentile
is a value x_p of x such that p% of the observed values of x are less than x_p.

How to calculate the Percentile:
1. Count the total item in dataset = N
2. Multiply the percentile p with total no of items = N*p
3. This will give you a no which can be a float or integer
4. If it is a float, round off it to nearest integer, named p^th no
          i. Sort the data into increasing order
          ii. Now, p^th no in this dataset is your percentile value
5. If it is an integer no
          i. Sort the data into increasing order
          ii. Now, average of p^th no and (p+1)^th no in this dataset is your percentile value

So when we say, 20% means -
No of items in dataset = 9
No of items which should be less than x_p. - 9*20% = 1.8
Round off this to nearest integer - 2
Our dataset is already sorted in increasing order, so check the 2nd value - 12

likewise, 25%, 50% and 75% is - 9*25%, 9*50%, 9*75% = 2.25^th, 4.5^th, 6.75^th
2^th, 5^th, 7^th - 12, 23, 45

This is one way to calculate the percentile, If you use calculator or some other method, it might be slightly different.

Mean or Average:  Sum(all items) / Total no of element

Mean -  (9+12+12+17+23+43+45+67+91)/9 = 34.4

However, the mean is very sensitive to outliers. So to understand the data tendency, we go for median rather than means.

Median: Median is 50 percentile, or middle value

How to get Median/Middle value - a. Sort the data into increasing orderb. Get total no of elements - N     if N is even -  median =   ( N/2th element + [N/2 + 1]th element) / 2     if N is odd - median = ceil(N/2)th element

For our case, N = 9, which is odd, so ceil(9/2) = ceil(4.5) = 5th element 

Median = 23

Range:  Difference between Max and Min is called range. 

Input dataset range - 91-9 = 82

Variance: The variance or standard deviation is the most common measure of the spread of a set of points.

`variance(x) = \sigma^2 = \frac{1}{n-1}\Sigma_{i=1}^n(x_i-\bar{x})^2`
where `\bar{x}` is Mean of all value of x
m = total no of items in dataset
`\sigma` is standard deviation

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