Types of Averages (Means)
Posted on Fri 21 August 2020 in Data Science • 4 min read
The most common analytical task is to take a bunch of numbers in dataset and summarise it with fewer numbers, preferably a single number. Enter the 'average', sum all the numbers and divide by the count of the numbers. In mathematical terms this is known as the 'arithmetic mean', and doesn't always summarise a dataset correctly. This post looks into the other types of ways that we can summarise a dataset.
The proper term for this method of summarising is determining the central tendency of the dataset.
Generate The Data¶
First step is to generate a dataset to summarise, to do this we use the random package from the standard library. Using matplotlib we can plot our 'number line'.
import random
import typing
random.seed(42)
dataset: typing.List = []
for _ in range(50):
dataset.append(random.randint(1,100))
print(dataset)
import matplotlib.pyplot as plt
def plot_1d_data(arr:typing.List, val:float, **kwargs):
constant_list = [val for _ in range(len(arr))]
plt.plot(arr, constant_list, 'x', **kwargs)
plot_1d_data(dataset,5)
Median¶
The median is the middle number of the sorted list, in the quite literal sense. For example the median of 1,2,3,4,5 is 3; as is the same for 3,2,4,1,5. The median can be more descriptive of the dataset over the arithmetic mean whenever there are significant outliers in the data that skew the arithmetic mean.
If there is an even amount of numbers in the data, the median becomes the arithmetic mean of the two middle numbers. For example, the median for 1,2,3,4,5,6 is 3.5 (3+4/2).
When to use¶
Use the median whenever there is a large spread of numbers across the domain
import statistics
print(f"Median: {statistics.median(dataset)}")
plot_1d_data(dataset,5)
plt.plot(statistics.median(dataset),5,'x',color='red',markersize=50)
plt.annotate('Median',(statistics.median(dataset),5),(statistics.median(dataset),5.1),arrowprops={'width':0.1})
Mode¶
The mode of a dataset is the number the appears most in the dataset. It is to be noted that this is the least used method of demonstrating central tendency.
When to use¶
Mode is best used with nominal data, meaning if the data you are trying to summarise has no quantitative metrics behind it, then mode would be useful. Eg, if you are looking through textual data, finding the most used word is a significant way of summarising the data.
import statistics
print(f"Mode: {statistics.mode(dataset)}")
plot_1d_data(dataset,5)
plt.plot(statistics.mode(dataset),5,'x',color='red',markersize=50)
plt.annotate('Mode',(statistics.mode(dataset),5),(statistics.mode(dataset),5.1),arrowprops={'width':0.1})
Arithmetic Mean¶
This is the most used way of representing central tendency. It is done by summing all the points in the dataset, and then dividing by the number of points (to scale back into the original domain). This is the best way of representing central tendency if the data does not containing outliers that will skew the outcome (which can be overcome by normalisation).
When to use¶
If the dataset is normally distributed, this is the ideal measure.
def arithmetic_mean(dataset: typing.List):
return sum(dataset) / len(dataset)
print(f"Arithmetic Mean: {arithmetic_mean(dataset)}")
plot_1d_data(dataset,5)
plt.plot(arithmetic_mean(dataset),5,'x',color='red',markersize=50)
plt.annotate('Arithmetic Mean',(arithmetic_mean(dataset),5),(arithmetic_mean(dataset),5.1),arrowprops={'width':0.1})
Geometric Mean¶
The geometric mean is calculated by multiplying all numbers in a set, and then calculating the nth root of the multiplied figure, when n is the count of numbers. Since this using the multiplicative nature of the dataset to find a figure to summarise by, rather than an additive figure of the arithmetic mean, thus making it more suitable for datasets with a multiplicative relationship.
We calculate the nth root by raising to the power of the reciprocal.
When to use¶
If the dataset has a multiplicative nature (eg, growth in population, interest rates, etc), then geometric mean will be a more suitable way of summarising the dataset. The geometric mean is also useful when trying to summarise data with differenting scales or units as the geometric mean is technically unitless.
def multiply_list(dataset:typing.List) :
# Multiply elements one by one
result = 1
for x in dataset:
result = result * x
return result
def geometric_mean(dataset:typing.List):
if 0 in dataset:
dataset = [x + 1 for x in dataset]
return multiply_list(dataset)**(1/len(dataset))
print(f"Geometric Mean: {geometric_mean(dataset)}")
plot_1d_data(dataset,5)
plt.plot(geometric_mean(dataset),5,'x',color='red',markersize=50)
plt.annotate('Geometric Mean',(geometric_mean(dataset),5),(geometric_mean(dataset),5.1),arrowprops={'width':0.1})
Harmonic Mean¶
Harmonic mean is calculated by:
- taking the reciprocal of all the numbers in the set
- calculating the arithmetic mean of this reciprocal set
- taking the reciprocal of the calculated mean
When to use¶
The harmonic mean is very useful when trying to summarise datasets that are in rates or ratios. For example if you were trying to determine the average rate of travel over a trip with many legs.
def reciprocal_list(dataset:typing.List):
reciprocal_list = []
for x in dataset:
reciprocal_list.append(1/x)
return reciprocal_list
def harmonic_mean(dataset:typing.List):
return 1/arithmetic_mean(reciprocal_list(dataset))
print(f"Harmonic Mean: {harmonic_mean(dataset)}")
plot_1d_data(dataset,5)
plt.plot(harmonic_mean(dataset),5,'x',color='red',markersize=50)
plt.annotate('Harmonic Mean',(harmonic_mean(dataset),5),(harmonic_mean(dataset),5.1),arrowprops={'width':0.1})
print(f"Mode: {statistics.mode(dataset)}")
print(f"Median: {statistics.median(dataset)}")
print(f"Arithmetic Mean: {arithmetic_mean(dataset)}")
print(f"Geometric Mean: {geometric_mean(dataset)}")
print(f"Harmonic Mean: {harmonic_mean(dataset)}")
Thank you to Andrew Goodwin over on Twitter: https://twitter.com/ndrewg/status/1296773835585236997 for suggesting some extremely interesting further reading on Anscombe's Quartet and The Datasaurus Dozen, which are examples of why summary statistics matter of exactly the meaning of this post!