Descriptive Statistics

## Uncomment and run this cell to install the packages
# !pip install --upgrade dataidea

Descriptive Statistics and Summary Metrics

In this notebook, we will learn to obtain important values that describe our data including:

• Measures of central tendency
• Measures of variability
• Measures of distribution shape
• Measures of association

import pandas as pd
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

This notebook has been modified to use the Nobel Price Laureates Dataset which you can download from opendatasoft

# load the dataset (modify the path to point to your copy of the dataset)
data = pd.read_csv('../assets/nobel_prize_year.csv')
data = data[data.Gender != 'org'] # removing organizations
data.sample(n=5)

	Year	Gender	Category	birth_year	age
505	2014	male	Physics	1929	85
318	1952	male	Literature	1885	67
883	1933	male	Literature	1870	63
481	1995	male	Peace	1908	87
769	2005	male	Peace	1942	63

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What is Descriptive Statistics

Descriptive statistics is a branch of statistics that deals with the presentation and summary of data in a meaningful and informative way. Its primary goal is to describe and summarize the main features of a dataset.

Commonly used measures in descriptive statistics include:

1. Measures of central tendency: These describe the center or average of a dataset and include metrics like mean, median, and mode.

2. Measures of variability: These indicate the spread or dispersion of the data and include metrics like range, variance, and standard deviation.

3. Measures of distribution shape: These describe the distribution of data points and include metrics like skewness and kurtosis.

4. Measures of association: These quantify the relationship between variables and include correlation coefficients.

Descriptive statistics provide simple summaries about the sample and the observations that have been made.

1. Measures of central tendency ie Mean, Median, Mode:

The Center of the Data:

The center of the data is where most of the values are concentrated.

Mean: It is the average value of a dataset calculated by summing all values(numerical) and dividing by the total count.

mean_value = np.mean(data.age)
print("Mean:", mean_value)

Mean: 60.21383647798742

Median: It is the middle value of a dataset when arranged in ascending order. If there is an even number of observations, the median is the average of the two middle values.

median_value = np.median(data.age)
print("Median:", median_value)

Median: 60.0

Mode: It is the value that appears most frequently in a dataset.

mode_value = sp.stats.mode(data.age)[0]
print("Mode:", mode_value)

Mode: 56

Homework:

Other ways to find mode (ie using pandas and numpy)

2. Measures of variability

The Variation of the Data:

The variation of the data is how spread out the data are around the center.

a) Variance and Standard Deviation:

Variance: It measures the spread of the data points around the mean.

# how to implement the variance and standard deviation using numpy
variance_value = np.var(data.age)
print("Variance:", variance_value)

Variance: 159.28551085795658

Standard Deviation: It is the square root of the variance, providing a measure of the average distance between each data point and the mean.

std_deviation_value = np.std(data.age)
print("Standard Deviation:", std_deviation_value)

Standard Deviation: 12.620836377116873

Summary

In summary, variance provides a measure of dispersion in squared units, while standard deviation provides a measure of dispersion in the original units of the data

Note!

Smaller variances and standard deviation values mean that the data has values similar to each other and closer to the mean and the vice versa is true

plt.hist(x=data.age, bins=20, edgecolor='black')
# add standard deviation lines
plt.axvline(mean_value, color='red', linestyle='--', label='Mean')
plt.axvline(mean_value+std_deviation_value, color='orange', linestyle='--', label='1st std Dev')
plt.axvline(mean_value-std_deviation_value, color='orange', linestyle='--')
plt.title('Age of Nobel Prize Winners')
plt.ylabel('Frequency')
plt.xlabel('Age')
# Adjust the position of the legend
plt.legend(loc='upper left')

plt.show()

b) Range and Interquartile Range (IQR):

Range: It is the difference between the maximum and minimum values in a dataset. It is simplest measure of variation

# One way to obtain range
min_age = min(data.age)
max_age = max(data.age)
age_range = max_age - min_age
print('Range:', age_range)

Range: 80

# Calculating the range using numpy
range_value = np.ptp(data.age)
print("Range:", range_value)

Range: 80

Interquartile Range (IQR): It is the range between the first quartile (25th percentile) and the third quartile (75th percentile) of the dataset.

Quartiles:

Calculating Quartiles

The quartiles (Q0,Q1,Q2,Q3,Q4) are the values that separate each quarter.

Between Q0 and Q1 are the 25% lowest values in the data. Between Q1 and Q2 are the next 25%. And so on.

• Q0 is the smallest value in the data.
• Q1 is the value separating the first quarter from the second quarter of the data.
• Q2 is the middle value (median), separating the bottom from the top half.
• Q3 is the value separating the third quarter from the fourth quarter
• Q4 is the largest value in the data.

# Calculate the quartile
quartiles = np.quantile(a=data.age, q=[0, 0.25, 0.5, 0.75, 1])

print('Quartiles:', quartiles)

Quartiles: [17. 51. 60. 69. 97.]

Percentiles:

Percentiles are values that separate the data into 100 equal parts.

For example, The 95th percentile separates the lowest 95% of the values from the top 5%

• The 25th percentile (P25%) is the same as the first quartile (Q1).
• The 50th percentile (P50%) is the same as the second quartile (Q2) and the median.
• The 75th percentile (P75%) is the same as the third quartile (Q3)

Calculating Percentiles with Python

To get all the percentile values, we can use np.percentile() method and pass in the data, and the list of the percentiles as showed below.

# Getting many percentiles
percentiles = np.percentile(data.age, [25, 50, 75])
print(f'Percentiles: {percentiles}')

Percentiles: [51. 60. 69.]

To get a single percentile value, we can again use the np.percentile() method and pass in the data, and a the specicific percentile you're interested in eg:

# Getting one percentile at a time
first_quartile = np.percentile(a=data.age, q=25) # 25th percentile
middle_percentile = np.percentile(data.age, 50)
third_quartile = np.percentile(data.age, 75) # 75th percentile

print('Q1: ', first_quartile)
print('Q2: ', middle_percentile)  
print('Q3: ', third_quartile)

Q1:  51.0
Q2:  60.0
Q3:  69.0

Note!

Note also that we can be able to use the `np.quantile()` method to calculate the percentiles which makes logical sense as all the values mark a fraction(percentage) of the data

percentiles = np.quantile(a=data.age, q=[0.25, 0.50, 0.75])
print('Percentiles:', percentiles)

Percentiles: [51. 60. 69.]

Now we can be able to obtain the interquartile range as the difference between the third and first quartiles as predefined.

# obtain the interquartile
iqr_value = third_quartile - first_quartile
print('Interquartile range: ', iqr_value)

Interquartile range:  18.0

Note: Quartiles and percentiles are both types of quantiles

Summary

While the range gives an overview of the entire spread of the data from lowest to highest, the interquartile range focuses s`pecifically on the spread of the middle portion of the data, making it more robust against outliers.

Frequency Tables

Frequency means the number of times a value appears in the data. A table can quickly show us how many times each value appears. If the data has many different values, it is easier to use intervals of values to present them in a table.

Here's the age of the 934 Nobel Prize winners up until the year 2020. IN the table, each row is an age interval of 10 years

Age Interval	Frequency
10-19	1
20-29	2
30-39	48
40-49	158
50-59	236
60-69	262
70-79	174
80-89	50
90-99	3

Note: The intervals for the values are also called bin

Further Reading

Chapter 3 of An Introduction to Statistical Methods and Data Analysis 7th Edition_New

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