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Data Science

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.

Types_of_measures_of_central_tendency.png

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()

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()

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 {1, 2, 2, 3, 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