Statistical estimates of random variables

Measures of the central tendency

Mean

The following data-set is given: [71, 50, 48, 67, 53]

The mean of the given data set can be calculated by the following code:

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.mean()
print("Mean is :", x)

Mean is : 57.8

Weighted Mean

Another way to describe the data set [71, 48, 48, 67, 53] is to put it in to a table form:

f	f(x)
71	1
48	2
67	1
53	1

The weighted Mean can be calculated by the following code:

import pandas as pd
import numpy as np

ft = pd.DataFrame.from_dict({
    'x': [71, 48, 67, 53],
    'fx': [1, 2, 1, 1]
})

x = np.average(a = ft['x'], weights = ft['fx'])
print("Weighted mean is :", x)

Weighted mean is : 57.4

Median

The following data set is given: [71, 50, 48, 67, 53]

The middle-most value is the median and it can be calculated by the following code:

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.median()
print("Median is :", x)

Median is : 53.0

The median of the data set [48, 50, 53, 65, 67, 71] is:

import pandas as pd

dataset = pd.Series([48, 50, 53, 65, 67, 71])
x = dataset.median()
print("Median is :", x)

Median is : 59.0

Mode

The mode is the value that occurs most frequently.

The mode of the data set [71, 50, 48, 48, 53] can be calculated by the following code:

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.mode()
print("Mode is:", x[0])

Mode is: 48

If the values in a distribution have equal number of occurrences, then there isn’t a single mode.

The set of scores can be bimodal (with two modes), as the following set: [71, 48, 48, 50, 67, 67, 53]

import pandas as pd

dataset = pd.Series([71, 48, 48, 50, 67, 67, 53])
x = dataset.mode()
print("Modes are:", x[0], ",", x[1])

Modes are: 48 , 67

Measures of Variability

Range

The range is calculated by subtracting the lowest value in a data set from the highest value in the data set.

For the dataset [71, 50, 48, 67, 53] the range is:

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.max() - dataset.min()
print("Range is:", x)

Range is: 23

Standard Deviation

The standard deviation represents the average amount of variability in a data set

The standard deviation of the data set [71, 50, 48, 67, 53] is:

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.std()
print("Standard deviation is:", x)

Standard deviation is: 10.473776778220929

Variance

The variance, is simply the squared standard deviation and for the data set [71, 50, 48, 67, 53] is

import pandas as pd

dataset = pd.Series([71, 50, 48, 67, 53])
x = dataset.var()
print("Variance is:", x)

Variance is: 109.69999999999999

Coefficient of Variation

The coefficient of variation is a derivative of two other statistical measures - the standard deviation and the mean and it is the ratio between them

Two datasets are given: [71, 50, 48, 67, 53] and [171, 150, 148, 167, 153]. Their coefficients of variation are:

import pandas as pd

def Variation(values):
    return values.std() / values.mean()

dataset1 = pd.Series([71, 50, 48, 67, 53])
dataset2 = pd.Series([171, 150, 148, 167, 153])

x1 = Variation(dataset1)
x2 = Variation(dataset2)

print("Coefficient of variation:", x1)
print("Coefficient of variation:", x2)

Coefficient of variation: 0.18120721069586382
Coefficient of variation: 0.06637374384170425