Distribution of Order Statistics and Range

Statistical data needs to be organized and properly dealt with for accurate results. Scientists and students often face problems that need to be solved using basic statistical formulas and concepts. Here, we would be dealing with order statistics and other basic concepts that would surely help you in your studies and project. Read on to know more about how statistics can help you read and understand your data easily.

Order Statistics: An Introduction

In the field of statistical sciences, order statistics is an extremely important concept. Besides rank statistics, it is one of the most basic tools in non-parametric statistics and inference. For any given set of data, the kth order statistics is equal to the kth smallest value. Let us look at an example of order statistics in a random dataset of 10 numbers. 

Let there be a dataset A = {3, 5, 11, 2, 98, 41, 33, 298, 54, 63} 

This is a disorganized dataset, where the kth order statistics can be given as follows. 

x(1)=2 x(2)=3 x(3)=5 x(4)=11

x(5)=33 x(6)=41 x(7)=54 x(8)=63

x(9)=97 x(10)=298

The first order statistics is always the minimum value of the data presented.

Correlation Coefficient Range Formulas

Correlation coefficient is a value that is used to measure the relationship between two variables. The correlation coefficient formula can be given as:

xy=Cov(x,y)xy

In this formula, xy refers to the Pearson product-moment correlation coefficient

Cov(x,y) refers to the covariance of variables x and y.

x and y are the standard deviation of x and y respectively.

Covariance is given by Cov(x,y) = E{(X-EX)(Y-EY)] = E[XY]-(EX)(EY)

Standard deviation is given by s = (X-x)2n-1, 

Where s is the standard deviation, n is the total number of observations, X is the value in the dataset and x is the mean. 

For two variables, the correlation coefficient range is -1.0 to +1.0. This means that only values between -1.0 to +1.0 are valid. Value lies outside this range can mean that there has been a calculation error. 

Mean Median Mode Range Formulas

Mean refers to the average of all the values in a dataset.

Median refers to the value at the midpoint of an organized dataset. For a dataset with an even number of elements, it is the arithmetic mean of the two middle values.

For any dataset, mode refers to the element that appears the highest number of times. A dataset can have 0, 1 or more modes.

The statistics range of a dataset refers to the difference between the highest and lowest element. 

The mean, median, mode range formulas are given below. 

Mean = Sum of all elements of a datasetTotal number of elements in a dataset

Median ={n2th element + (n2+1)th element2    for even number of elementsn+12th element                          for odd number of elements

Range = Highest element-Lowest element

Here are a few mean median mode range examples to simplify your understanding of the topic. 

Example 1:

Sample dataset: {3, 5, 11, 2, 98}

Mean = 3+5+11+2+985 = 1195 = 23.8

Median:

Ordered dataset: {2,3,5,11,98}

Number of terms = n = 5 (odd number of elements)

So, median = n+12th term = 5-12the term = 3rd term = 5

Mode: This data set has no mode.

Range = 98-2 = 96

Example 2:

Sample dataset: {3, 72,  511, 2, 72, 28, 51, 72}

Mean = 3+72+511+2+72+28+51+728 = 8118 = 101.375

Median:

Ordered dataset: {2,3,28,51,72,72,72,511}

Number of terms = n = 8 (even number of elements)

So, median is given by n2th element + (n2+1)th element2=4th + 5th element2=51+722=1232=61.5

Mode: 72, appears 3 times

Range = 511-2 = 509

Conclusion

With this knowledge of the topic, you can solve several statistical data analysis related problems. You can try out the formulas mentioned here using sample datasets available online. Practice would help you understand the concepts more and make your application of the formulas fluid and simple.