Chapter 2: Description of Samples and Populations

STAT 218

Let’s Refresh Our Memory

Until now, we have:

  • identified the differences between anecdotal and empirical evidence
  • discussed the differences between observational and experimental studies

And OF COURSE

  • We took our first steps in RStudio!

What about Today?

Today we will:

  • have an introduction to descriptive statistics
    • the principles of data visualization (will continue next week with R)
    • see examples of frequency distributions enabling us to summarize a categorical variable or a quantitative variable
      • either as a table or a graph
  • see how to identify/compute shape, center, and dispersion of a quantitative variable.

Making Sense of Data


Data sets can be messy and overwhelming at first glance. By using descriptive statistics, we can:

  • Visualize to reveal patterns,
  • Summarize key information, and
  • Organize data for better understanding.

Data Visualization

Introduction


  • One way of representing data (graphical)

    • Other ways are data sonification and data in tactile format.

  • Infographics, charts, plots and even maps are examples of visualizing data.

    • In this class, we will use the coordinate system, different colors, and shapes to visualize data.

  • They can tell a story or help explain/explore data.

A Successful Data Visualization…

According to Royal Statistical Society, a successful data visualization will:

  • Grab attention
  • Improve access to information
  • Increase precision
  • Bolster credibility
  • Summarise content

Examples


The Paintings of Bob Ross

Mandatory Paid Vacation

Why are K-pop groups so big?

History of Philosophy

Leonardo DiCaprio Refuses to Date a Woman Over 25

Further Resources

Vocabulary Before We Start

Population and Sample

  • We’d like to know about an entire collection of individuals, called a population, but examining all of them is usually impractical, if not impossible. (e.g., all students in Cal Poly)

  • So we select a smaller group of individuals, a sample, from the population.

    • e.g. 1500 (n = 1500) students in Cal Poly.

  • A subset of a larger population that accurately reflects the characteristics of the whole group is said to be representative sample.

  • To make the sample as representative as possible, select individuals for the sample by employing randomness.

Vocabulary Time!

Descriptive Statistics: Statistics used for describing and summarizing data.

Parameter: A number represents the entire population (e.g., population mean).

Statistic: A number calculated from a sample (e.g., sample mean).

Frequency Distributions

Introduction

  • A frequency distribution is a representation of the frequency, indicating how often each value appears in a data set.
  • This information can be conveyed through tables or, more visually, using a graph.

Summarizing A Categorical Variable

A Frequency Distribution Table

In this section, you will see two examples for summarizing a single categorical variable.


Frequency Distribution of Smoking Status
Smoking_Status Frequency
FALSE 742
TRUE 484
Total 1226

Bar Chart

A visual representation of categorical data showing the number of observations in each category.

Another Example

Climate change affects the quality of human life
Likert_Response Frequency
Strongly Disagree 13
Disagree 21
Neutral 18
Agree 23
Strongly Agree 25
Total 100

Summarizing A Quantitative Variable

Grouped Frequency Distributions

Histogram

  • A visual summary of a quantitative variable.

    • The x-axis represents the range of values divided into bins,
    • The y-axis represents the count of data points in each bin.

  • Histograms help identify shape of the distribution, outliers, central tendency, and variability.

  • No Gaps between Bars: Unlike a bar chart, there are no gaps between the bars in a histogram because the bins are continuous.

More on Histogram


What’s so hard about histograms?

Shapes of Distributions

A Quick Snapshot

Shapes of Distributions

The shape of a distribution can be represented by a smooth curve approximating of the histogram.

Central Tendency

Measures of Center

To understand the center or typical value of a data set, we calculate

  • Mean OR
  • Median

Mean

  • You might be familiar with this term. It is also known as
    • arithmetic mean OR sample mean

Tip

We employed a symbolic convention to differentiate between a variable and an observed value of that variable.

  • \(Y = birthweight\) (Variable)
  • \(y = 12.8\) lb (Observed Value)

We now denote

  • the observations in a sample by \(y_1\), \(y_2\), . . . , \(y_n\)
  • the mean of the sample by the symbol \(\bar{y}\) (read “y-bar”).

Mean

We calculate the mean by using this formula

Median

  • Imagine what would happen if Bill Gates was in our class and we calculated the average money in our bank account.
    • It might not be the best idea to interpret this average.
  • Instead, we can calculate median which is a value that splits the ordered data into two equal parts.

How to Find the Median

  • Arrange the observations in increasing order.
  • In the array of ordered observations, the median is
    • the middle value (if n is odd) or
    • midway between the two middle values (if n is even).
  • We denote the median of the sample by the symbol \(\tilde{y}\) (read “y-tilde”).

Let’s have another toy example

  • Assume that we have a following dataset:
    • 22, 6, 6, 4, 2
Measures of Center Data and Calculation Result
Mean (2+4+6+6+22)/5 8
Median 2,4,6,6,22 6

Understanding Mean and Median

Spread of Distributions

Let’s assume we managed to collect data from our squirrels on campus :) Our class was divided into three groups, and each group measured the weights (lbs) of 10 squirrels. Here are the results:

Group 1: 1.25, 1.25, 1.25, 1.25, 1.25, 1.25, 1.25, 1.25, 1.25, 1.25

Group 2: 1.0, 1.0, 1.0, 1.0, 1.0, 1.5, 1.5, 1.5, 1.5, 1.5

Group 3: 1.0, 1.4, 1.2, 1.4, 1.1, 1.3, 1.6, 1.0, 1.2, 1.3

Dr. Demirci mentioned that looking at these numbers is so confusing. Can you please calculate the sample mean for them to summarize this data?

All these groups calculated the same mean, which is 1.25 lbs. Dr. Demirci seemed not so happy with this number.

  • Why?