Measures of Central Tendency

Measures of central tendency

Measures of central tendency are statistical tools used to identify the central point or typical value of a dataset.

Type of Measures of Central tendency

1. Mean (Arithmetic Mean).
2. Median.
3. Mode.

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Practical use cases of Central Tendency

• EDA.
• Handling Missing Values.
• Feature Engineering.
• Normalization and Standardization.
• Data Distribution Analysis.

Mean (Arithmetic mean)

definition of mean

The sum of all values divided by the number of values.

Formula of mean

$$\text{Mean} = \frac{x_1 + x_2 + \dots + x_n}{n}$$

Example of mean

• For the dataset [5, 10, 15], the mean is

$$\frac{5+10+15}{3} = \frac{30}{3}=10$$

Advantage of mean

• Provides a single value that summarizes the entire dataset.
• Useful when all values in the dataset are equally important.

Disadvantage of mean

• Sensitive to outliers: If the dataset contains very large or very small values (outliers), the mean can be skewed, making it an unreliable measure of central tendency for such data.

Numpy in Mean

• How to find mean in numpy

import numpy as np

array = np.array([1,2,3,4])

print('mean = ', np.mean(array)) # output 2.5

# other way to find mean

print('mean = ', array.mean()) # output 2.5

Pandas in Mean

• How to find mean in pandas

import pandas as pd

df = pd.DataFrame({'A': [1, 2, 3], 'B': [4, 5, 6]})

# Mean of all numeric columns
df.mean() 

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Median

definition of Median

The middle value in a dataset when the values are arranged in ascending or descending order.

Formula of Median

Odd number of data points

$$\text{Median} = \left( \frac{n + 1}{2} \right)\text{th value}$$

Even number of data points

$$\text{Median} = \frac{\left( \frac{n}{2} \text{th value} \right) + \left( \frac{n}{2} + 1 \text{th value} \right)}{2}$$

Example of Median

• Let's say we have the following set of numbers: 2, 5, 8, 11, 15.

To find the mean:

$$\text{Step 1: } 2 + 5 + 8 + 11 + 15 = 41$$ $$\text{Step 2: } \frac{41}{5} = 8.2$$ $$\text{Therefore, the Mean = 8.2}$$

Advantage of Median

• Resistant to outliers.
• Works well for skewed distributions.
• Applicable for ordinal data.
• Clear measure of central location.
• Simple to calculate.

Disadvantage of Median

• Ignores data distribution.
• Less informative for symmetric distributions.
• Cannot be used for further mathematical operations.
• Sensitive to sampling.
• Less stable for grouped data.

Numpy in Median

• How to find Median in numpy

arr = np.array([1,2,3,4,20])

print('median = ',np.median(arr)) #output 3

Pandas in Median

• How to find median in pandas

import pandas as pd

df = pd.DataFrame({'A': [1, 2, 3], 'B': [4, 5, 6]})

# median of all numeric columns
df.median() 

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Mode

definition of Mode

The mode is the value that appears most frequently in a dataset.

Formula of Mode

• sort the give Value.

• find the most frequently number and this number is mode.

Example of Mode

• In the dataset 4, the mode is 2 because it appears more times than any other number.

Advantage of Mode

• Simple to understand.
• Useful for categorical data.
• Not affected by outliers.
• Applicable to non-numerical data.
• Can indicate multiple modes (multimodal).
• Does not require full dataset knowledge.

Disadvantage of Mode

• Not always unique or well-defined.
• Less stable for small datasets.
• May not represent central tendency well.
• Ignores much of the data.
• Difficult to use with continuous data.
• Not suitable for advanced analysis.

Numpy in Mode

• How to find meaModen in numpy

from scipy import stats
import numpy as np
arr = np.array([1,2,3,4,3,20])

print('Mode = ',stats.mode(arr)) #output 3

Pandas in Mode

• How to find Mode in pandas

import pandas as pd

series = pd.Series([1, 2, 2, 3, 3, 3, 4])
mode_value = series.mode()
print(mode_value)  # Output: 3

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