Descriptive Statistics

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Contributor:

Erica Madebeykin
PGDip Researcher

Creation Date: March 12, 2024

❤️ Reference Acknowledgments: University of Amsterdam (world rank #53)

This review reflects the author's learning and thoughts based on the materials covered in the FREE online course provided by the University of Amsterdam, delivered on the coursera platform.
A heartfelt gratitude to Professors Emiel van Loon and Matthijs Rooduijn for delivering such an amazing course!

Progressive pattern of formulas

Formula Patterns

As you learn about descriptive statistics, you'll see that each step has a progressive pattern: from finding the mean to doing linear regressions. Each formula guides you to the next one, making the process clear and connected.

For example, when you understand how to find the mean, it leads you to other analyses like Pearson's r and linear regressions, which help you understand relationships between things you're studying. Another example is the median, which leads you to the IQR.
On this page, we will decode these patterns and begin a journey of joyful learning!

First Pattern: Mean to Linear Regression progressive formulas
You will learn the following:

• overview of formula patterns: Mean, Standard Deviation, Z-scores, Pearson's R, Linear Regression, residual errors,etc.
• why and when do we use these formulas
• applied statistical formulas in real business cases
  

Second Pattern: Median to IQR formulas
You will learn the following:

• overview of formula patterns: Median, IQR, Q1, Q2, Q3,etc.
• why and when do we use these formulas
• applied statistical formulas in real business cases

First Set of Patterns: from Mean Formula to Linear Regression Coefficients

Mean Formula

• simply the average of values

$$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}$$

Standard Deviation Formula

$$s_x=\sqrt{\frac{\sum _{i=1}^n(x_i-\overline{x}{)}^2}{n-1}}$$

Z- Score Formulas

x values

• How to get the Z-score of x values:

• x-value minus x-bar ; divided by the standard deviation of x

  $$z_{x_i}=\frac{x_i-\overline{x}}{s_x}$$

• s is the standard deviation of x

y values

• How to get the Z-score of y values:

• y-value minus y-bar ; divided by the standard deviation of y-variable

  $$z_{y_i}=\frac{y_i-\overline{y}}{s_y}$$

• 's' here is the standard deviation of y-variable!

Pearson's r Formula

Original formula (with separate values of x and y)

• Important: for Pearson's r, standard deviations are calculated separately for x- variable and another standard deviation for y-variable

$$r=\frac{\sum _{i=1}^n\left(\frac{x_i-\overline{x}}{s_x}\right)\cdot \left(\frac{y_i-\overline{y}}{s_y}\right)}{n-1}$$

Step-by-Step Calculation Breakdown for Understanding Pearson's r Formula

• 1st step: Get separately the x-bar (mean for x-values) and y-bar (mean for y-values)

• 2nd step: Get separately the corresponding standard deviations of x-values and then the y-values standard deviation

• 3rd step: Get every Z-score for each x value.

  $$Z_{x_i}=\frac{x_i-\overline{x}}{s_x}$$

• Note: For the denominator, make sure you use the standard deviation formula for x-variable

• 4th step: Get every Z-score for each y value :

  $$Z_{y_i}=\frac{y_i-\overline{y}}{s_y}$$

• Note: For the denominator, make sure you use the standard deviation formula for y-variable

• 5th step: Multiply every Z-score of X_i by each corresponding Z-score of Y_i

• example: Zx_1* Zy_1 ; Zx_2 * Zy_2 ; Zx_3 * Zy_3

  $$(Z_{x_1}\cdot Z_{y_1})=Z_{x{y}_1}$$$$(Z_{x_2}\cdot Z_{y_2})=Z_{x{y}_2}$$$$(Z_{x_3}\cdot Z_{y_3})=Z_{x{y}_3}$$

• 6th step: Get the sum of all products , which means adding the products like this

$$\sum Z_{x{y}_1}+Z_{x{y}_2}+Z_{x{y}_3}+...$$

• 7th step: Divide the sum by n-1

Simplified Formula

For quick memorization of the Pearson's r, here is the simplified formula from the University of Amsterdam:

$$r=\frac{\sum (Z_x{Z}_y)}{n-1}$$

Note: The author is currently in the process of writing, editing, and coding this article. Soon, topics like the applied statistical formulas in real business operations will be added to this page.