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Measures of Central Tendency

Measures of central tendency characterize a distribution by using an average value. These metrics describe what can be considered a "typical" or "representative" observation within the data set.

Mean

(for numeric attributes) The arithmetic mean \( \bar{x} \) is the most common and effective numerical measure for describing the average value of a distribution.

Definition: Arithmetic Mean
\[ \bar{x}=\frac{\sum_{i=1}^{N}x_i}{N}=\frac{x_1+x_2+\dots+x_N}{N} \]

with \( x_1, x_2, \dots, x_N \) representing a set of \( N \) values of a metric variable \( X \).

One disadvantage of the mean is its sensitivity to outliers. Even a small number of extreme values can distort the result.

import statistics 

statistics.mean([1,2,1,2,3,4,1,100,1,2,1])
>>> Output
10.73

To address this issue, the trimmed mean is used. This method involves removing a percentage of values from both the upper and lower ends of the distribution (e.g., 1%). 

from scipy import stats

#calculate 10% trimmed mean
stats.trim_mean([1,2,1,2,3,4,1,100,1,2,1], 0.1)
>>> Output
1.889

However, this approach can lead to a loss of information, especially when a large portion of the data is trimmed.

Example: Mean of the Temperature

Given a table with 14 temperature values in °C, the goal is to calculate the mean and the trimmed mean (20% left and right) of the distribution. 

[28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
Solution

\[ \begin{eqnarray*} \bar{x} &=& \frac{28.3 + 27.2 + 27.4 + 22.7 + 14.3 + 11.9 + 13.8}{14}\\ &&+\frac{19.8 + 9.6 + 21.1 + 20.8 + 19.8 + 25.3 + 22.8}{14} \\ &=& 20.34 \end{eqnarray*} \]
\[ \begin{eqnarray*} \bar{x}_{trim} &=& \frac{13.8+ 14.3+ 19.8+ 19.8+20.8 }{14}\\ &&+\frac{21.1+ 22.7+ 22.8+ 25.3+ 27.2}{14} \\ &=& 20.76 \end{eqnarray*} \]

The mean temperature is \(20.34^\circ C\) and the trimmed mean is \(20.76^\circ C\) .

Code 
import statistics 
Temp = [28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
print('Mean: ' + str(statistics.mean(Temp)))

from scipy import stats
print('Trimmed Mean: ' + str(stats.trim_mean(Temp, 0.2)))

Median

(for numeric and ordinal attributes) The median is more robust against outliers, as it splits the distribution into an upper and a lower half. It corresponds to the 50th percentile or the second quartile. 

import numpy as np
np.sort([1,2,1,2,3,4,1,100,1,2,1])
>>> Output
[1,1,1,1,1,2,2,2,3,4,100]
           ↑

statistics.median([1,2,1,2,3,4,1,100,1,2,1])
>>> Output
2

However, one downside is that it can be more complex to calculate, especially when dealing with large datasets.

Definition: Median

A data set of \( N \) values of an attribute \( X \) is sorted in increasing order

  • If \( N \) is odd, the median is the middle value of the ordered set
  • \( N \) is even, the median is the two middlemost values and any value in between (average of those two for numeric attribute )
Example: Median of the Temperature

Given a table with 14 temperature values in °C, the goal is to calculate the median of the distribution. 

[28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
Solution 
[9.6, 11.9, 13.8, 14.3, 19.8, 19.8, 20.8, 21.1, 22.7, 22.8, 25.3, 27.2, 27.4, 28.3]

\[ \bar{x} =\frac{20.8+21.1}{2} = 20.95\\ \]

The median temperature is \(20.95^\circ C\).

Code 
import statistics 
Temp = [28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
print('Median: ' + str(statistics.median(Temp)))

Mode

(for numeric, ordinal and nominal attributes) The mode is the most frequently occurring value in a distribution. It is also more robust against outliers, making it a useful measure in certain cases where extreme values might distort other central tendency metrics.

# Calculate the first Mode
statistics.mode([1,2,1,2,3,4,1,100,1,2,2])
>>> Output
1

# Calculate all Modes
statistics.multimode([1,2,1,2,3,4,1,100,1,2,2])
>>> Output
[1, 2]
Definition: Mode
  • Mode is the value that occurs most frequently in a data set.
  • There can be more than one mode (unimodal, bimodal, multimodal)
  • If each data value occurs only once, then there is no mode
Example: Mode of the Temperature

Given a table with 14 temperature values in °C, the goal is to calculate the mode of the distribution. 

[28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
Solution The mode of the temperature is \(19.8^\circ C\) because this value appears twice, making it the only value that occurs more than once. The distribution is unimodal.

Code 
import numpy as np
Temp = [28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]

from scipy import stats
#https://docs.scipy.org/doc/scipy/reference/stats.html
print('Mode: ' + str(stats.mode(Temp))) # returns the first mode (for unimodal)

import statistics 
#https://docs.python.org/3/library/statistics.html
print('Mode: ' + str(statistics.mode(Temp))) # returns the first mode (for unimodal)
print('Mode (Multi): ' + str(statistics.multimode(Temp))) # returns a list of all modes (for multimodal)

Quantile

(numeric and ordinal attributes) The \( q \)-quantile divides the data into \( q \) equal-sized parts. There are \( q - 1 \) quantiles (for example, 3 quantiles divide the data into 4 parts). The 2-quantile corresponds to the median. 

import numpy as np
np.sort([1,2,1,2,3,4,1,100,1,2,2])
>>> Output
[1,1,1,1,2,2,2,2,3,4,100]

The quartiles can be determined graphically using

[1,1,1,1,2,2,2,2,3,4,100]
 |----|----|----|----|
      Q1   Q2   Q3

or by using pythons scipy package

from scipy import stats

stats.mstats.mquantiles([1,2,1,2,3,4,1,100,1,2,2], prob=[0.25, 0.5, 0.75])
>>> Output
[1. , 2. , 2.8]

There are different methods for determining quantiles, especially when \( N \) is even. As a result, it is possible for different libraries to produce a different result than a manual calculation of quantiles.

Definition: Quantile

A data set of \(N\) values of an attribute \(X\) is sorted in increasing order

  • The \(k\)-th \(q\)-quantile is the value \(x\) where \(k/q\) of the data values are less and \((q-k)/q\) values are more than \(x\) (with \(0 < k < q\))
  • If set of numbers are odd, you have to calculate the middle
Image title

Most widely used forms

  • 2-quantile = median: Divides the data set in halves
  • 4-quantile = quartiles: Three data points split the data into four equal parts
  • 100-quantile = percentiles: Divide the data into 100 equal-sized sets
Example: Quantiles of Temperature Distribution

Given a table with 14 temperature values in °C, the goal is to calculate the quartiles of the distribution. 

[28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]
Solution 
[9.6, 11.9, 13.8, <<14.3>>, 19.8, 19.8, <<20.8>>, <<21.1>>, 22.7, 22.8, <<25.3>>, 27.2, 27.4, 28.3]

\[ Q_1 = 14.3^\circ C \qquad Q_2 = 20.95^\circ C \qquad Q_3 = 25.3^\circ C \]
Code 
from scipy import stats
#https://docs.scipy.org/doc/scipy/reference/stats.html

print('Quantile: ' + str(stats.mstats.mquantiles(Temp, prob=[0.25, 0.5, 0.75])))

Five Number Sumary & Boxplot

A single metric alone is not sufficient to fully characterize a distribution. The Five Number Summary is a useful approach to gain a deeper understanding of a distribution by combining multiple key metrics, offering a more comprehensive view of the data.

Definition: Five Number Summary

The Five Number Summary consists of the following metrics in this exact order:

[Minimum, \(Q_1\), Median, \(Q_3\), Maximum]

A boxplot is a visualization of a distribution and provides a graphical representation of the Five Number Summary. 

import plotly.express as px

fig = px.box([1,2,1,2,3,4,1,100,1,2,2])
fig.show()

The first and third quartiles (\(Q_1\) and \(Q_3\)) mark the ends of the box, while the interquartile range (IQR) represents the length of the box. The median is depicted as a line within the box. Two lines, known as whiskers, extend from the box to the smallest and largest observations, provided these values are no more than \(1.5\times\)IQR above \(Q_3\) or below \(Q_1\). Outliers are marked separately.

Example: Boxplot of the Temperature

  • Without Outlier

    Code 
    import plotly.express as px
import pandas as pd

Temp = [28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8]

# Generate Box Plot
fig = px.box(pd.DataFrame(Temp))

# Adjust the plot
fig.update_layout(
    xaxis_title_text=' ',
    yaxis_title_text='Temperature',
    title=dict(
            text='<b><span style="font-size: 10pt">Boxplot - no Outlier</span> <br> <span style="font-size:5">variable: Temperature</span></b>',
        ),
)

# Show the plot
fig.show()

  • With Outlier

    Code 
    import plotly.express as px
import pandas as pd

Temp2 = [28.3, 27.2, 27.4, 22.7, 14.3, 11.9, 13.8, 19.8, 9.6, 21.1, 20.8, 19.8, 25.3, 22.8, 50, 60]

# Generate Box Plot
fig = px.box(pd.DataFrame(Temp2))

# Adjust the plot
fig.update_layout(
    xaxis_title_text=' ',
    yaxis_title_text='Temperature',
    title=dict(
            text='<b><span style="font-size: 10pt">Boxplot - Outlier</span> <br> <span style="font-size:5">variable: Temperature</span></b>',
        ),
)

# Show the plot
fig.show()

Recap

  • Measures of central tendency characterize a distribution by using an average value.
  • Common metrics include the mean, median, mode, and quantiles.
  • Individual metrics alone are not sufficient to provide a full understanding.
  • The Five Number Summary combines multiple values to offer a more comprehensive characterization of the distribution.
  • The boxplot is the graphical representation of the Five Number Summary.
  • However, the informative value of the Five Number Summary is limited.
  • It can be misleading to describe a distribution solely through measures of central tendency.

The following distributions have the same mean, median and mode

  • Without Outlier

  • With Outlier

Code 
a = [2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4]
b = [1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5]

import plotly.express as px
import pandas as pd
from scipy import stats
import numpy as np

# Distribution a
# Generate Histogram
fig = px.histogram(pd.DataFrame(a))

# Adjust the plot
fig.update_layout(
    xaxis_title_text='Value',
    yaxis_title_text='Absolute Frequency',
    showlegend=False,
)

# Show the plot
fig.show()

# Calculate Measures
print('Mean: ' + str(np.mean(a)))
print('Median: ' + str(np.median(a)))
print('Mode: ' + str(stats.mode(a)))

# Distribution b
# Generate Histogram
fig = px.histogram(pd.DataFrame(b))

# Adjust the plot
fig.update_layout(
    xaxis_title_text='Value',
    yaxis_title_text='Absolute Frequency',
    showlegend=False,
)

# Show the plot
fig.show()

# Calculate Measures
print('Mean: ' + str(np.mean(b)))
print('Median: ' + str(np.median(b)))
print('Mode: ' + str(stats.mode(b)))

Tasks

Task: Measures of Central Tendency

Use the following dataset: 

from ucimlrepo import fetch_ucirepo 

# fetch dataset 
cars = fetch_ucirepo(id=9) 
# https://archive.ics.uci.edu/dataset/9/auto+mpg

# data (as pandas dataframes) 
data = cars.data.features
data = data.join(cars.data.ids)

# Show the first 5 rows
data.head()
Work on the following task:

  1. Calculate all useful and meaningful measure of central tendency (mean, median, mode) for the following attributes (think about attribute types)
    • car_name
    • origin
    • displacement
  2. Calculate the quartiles and plot a Boxplot fo the attribute acceleration