Measures of Variability (or Dispersion )

Definition

A measure of dispersion is a statistical measure that describes the spread or variability of a dataset. These measures help identify whether the data points are closely packed or widely spread around a central value (like the mean or median).

Type of Measures of Variability (or Dispersion )

1. Range.
2. Variance.
3. Standard Deviation.
4. Coefficient of Variation.

Practical use cases of Variability (or Dispersion )

• Assessing the spread in stock prices.
• Quality control in manufacturing.
• Risk assessment in finance (using STD).
• Outlier detection in data analysis.

Range (Arithmetic Range)

definition of Range

Range is the difference between the highest and lowest values in a dataset.

Formula of Range

• Range=Maximum value−Minimum value.

Example of Range

Advantage & Disadvantage of Range

Advantage of Range

• Useful for Small Data Sets.
• First Step in Identifying Outliers.
• Quick Insight into Data Spread.

Disadvantage of Range

• Sensitive to Outliers.
• Ignores Data Distribution.
• Provides No Information About Centrality.
• Not Robust for Large Datasets.

Practice of Range

Numpy in Range

• How to find Range in numpy

import numpy as np

data = np.array([10, 15, 20, 25, 30])

# Calculate range
range_value = np.max(data) - np.min(data)

print("Range using NumPy:", range_value)

Pandas in Mean

• How to find Range in pandas

import pandas as pd

data = pd.DataFrame({
    'A': [10, 15, 20, 25, 30],
    'B': [35, 45, 55, 65, 75]
})

# Calculate range for each column
range_values = data.max() - data.min()
print("Range for each column using Pandas:\n", range_values)

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Variance

Definition of Variance

Variance is a statistical measurement of the spread between numbers in a data.

info

Variance is often depicted by this symbol: σ2.

Formula of Variance

Population

Sample

Where

• σ ² = variance
• x _i = each individual data point
• μ = mean of the data points
• N = number of data points
• x̄ mean of the sample data points
• s ² = sample variance

Example of Variance

• Suppose we have a dataset of the ages of six people: 22,27,24,20,23,2622,27,24,20,23,26.

1. Calculate the Mean

• The mean (μ) of the data is:

2. Subtract the Mean and Square the Result.

For each value in the dataset, subtract the mean and then square the result.

3. Find the Average of the Squared Differences (Variance)

• For Population Variance

• For Sample Variance:

Advantage & Disadvantage Variance

Advantage of Variance

• Measures Data Spread
• Foundation for Other Metrics
• Highlights Data Variability
• Outlier Sensitivity

Disadvantage of Variance

• Units are Squared
• Sensitive to Outliers
• Difficult to Interpret
• Not Ideal for Non-Normal Distributions
• Complexity in Large Data

Practice of Variance

Numpy in Variance

• How to find Variance in numpy

import numpy as np

data = np.array([10, 15, 20, 25, 30])

print("Population variance :", np.var(data))
print("Sample variance :", np.var(data,ddof=1))

Pandas in Variance

• How to find Variance in pandas

import pandas as pd

df = pd.DataFrame({
    'A': [10, 15, 20, 25, 30],
    'B': [35, 45, 55, 65, 75]
})

print("Population variance :", df.var())
print("Sample variance :", df.var(ddof=1))

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Standard Deviation

Definition of Standard Deviation

a measure of the amount of variation or dispersion in a set of data values. It indicates how much individual data points deviate, on average, from the mean (average) of the dataset.

Formula of Standard Deviation

Population of SD

Sample of SD

Where of Formula

• σ ² = variance
• x _i = each individual data point
• μ = mean of the data points
• N = number of data points
• x̄ mean of the sample data points
• s ² = sample variance

Example of Standard Deviation

• Let's calculate the standard deviation for the dataset representing the ages of six people: 22,27,24,20,23,26.

1. Calculate the Mean

• The mean (μ) of the data is:

2. Subtract the Mean and Square the Result. For each value in the dataset, subtract the mean and then square the result.

3. Find the Average of the Squared Differences (Variance)

• For Population Variance

• For Sample Variance:

4. Take the Square Root to Get Standard Deviation

   1. For Population Standard Deviation (𝜎):

   2. For Sample Standard Deviation (𝜎):

Than Population Standard Deviation is 2.36 and Sample Standard Deviation is 2.58.

Practice of Standard Deviation

Numpy in Standard Deviation

• How to find Standard Deviation in numpy

import numpy as np

data = np.array([10, 15, 20, 25, 30])

print("Population Standard Deviation :", np.std(data))
print("Sample Standard Deviation :", np.std(data,ddof=1))

Pandas in Standard Deviation

• How to find Standard Deviation in pandas

import pandas as pd

df = pd.DataFrame({
    'A': [10, 15, 20, 25, 30],
    'B': [35, 45, 55, 65, 75]
})

print("Population Standard Deviation :", df.std())
print("Sample Standard Deviation :", df.std(ddof=1))

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Coefficient of Variation

Definition of Coefficient of Variation

Coefficient of Variation (CV) is a statistical measure that represents the ratio of the standard deviation to the mean. It expresses the extent of variability in relation to the mean of the dataset, allowing for comparison of relative variability between different datasets, regardless of their units or scales.

Formula of Coefficient of Variation

Where of Formula

• σ = population standard deviation
• μ = population mean

Example of Coefficient of Variation

• Let's calculate the standard deviation for the dataset representing the ages of six people: 22,27,24,20,23,26

• In before i have calculate standard deviation and Mean.Standard deviation = 2.6 and mean = 5.6.

• ans : Standard deviation = 2.6 / mean = 5.6 = 0.46.

Practice of Coefficient of Variation

Numpy in Coefficient of Variation

• How to find Coefficient of Variation in numpy

import numpy as np

data = np.array([10, 15, 20, 25, 30])

ans_population = np.std(data) / np.mean(data)
ans_Sample = np.std(data,ddof=1) / np.mean(data)

print("population Coefficient of Variation :", ans )
print("Sample Coefficient of Variation :", ans_Sample)

Pandas in Coefficient of Variation

• How to find Coefficient of Variation in pandas

import pandas as pd

df = pd.DataFrame({
    'A': [10, 15, 20, 25, 30],
    'B': [35, 45, 55, 65, 75]
})

cv = lambda x: np.std(x, ddof=1) / np.mean(x) 

print(df.apply(cv))

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