Analyzing Univariate Data

Introduction to Data Analysis

• Data analysis is defined as the process of collecting and organizing data in order to draw helpful conclusions from it.

• The primary purposes of analyzing data are:

  • To understand relationships and trends in the world around us.

  • To make predictions based on that information.

• Examples of applications for data analysis include:

  • Personal Health Data: Tracking the number of COVID-19 cases per day.

  • Weather Data: Monitoring daily temperature or the amount of rainfall.

  • Election Results: Polls tracking which candidates voters are likely to vote for.

  • Sports Data.

  • Census Data.

Methods of Data Collection

• Data can be collected from two primary sources:

  • Sample: A small group of people or items.

  • Census: An entire population.

• Data collection methods include:

  • Surveys.

  • Measurements.

  • Records.

  • Observations.

• Advancements in technology directly correlate with an increased ability to measure and store data.

Categorizing Data Analysis Approaches

• The approach to data analysis is determined by the number of variables involved in the data set.

Univariate (Single-Variable) Analysis

• Univariate sets provide measures of exactly one attribute.

• Example: The analysis of a set of test scores for a single class.

• Visualization tools for univariate analysis:

  • Bar graphs.

  • Pie graphs.

  • Boxplots.

  • Stem and leaf plots.

• Statistical tools for univariate analysis:

  • Mean.

  • Median.

  • Mode.

Bivariate (Two-Variable) Analysis

• Bivariate sets provide measures of two attributes for each specific item in a sample.

• Example: Comparing a student’s test score to the amount of time that student spent studying for the test.

• Visualization and trends in bivariate analysis:

  • Scatterplots: Used to see the relationship between variables.

  • Line of best fit or Curve of best fit: Used to represent the overall trend in the data.

Identification of Data Types (Example 1)

• Situation A: Noah researches how many cups of coffee Canadians drink.

  • Classification: Univariate data.

• Situation B: A study compared the length of time children spend playing video games and the time they spend reading.

  • Classification: Bivariate data.

Univariate Analysis: Measures of Central Tendency

• Measures of central tendency (mean, median, and mode) identify the center of a data set.

• Mean:

  • Definition: The mathematical average of a set.

  • Procedure: Add all the numbers in the set, then divide the resulting sum by the total number of items.

  • Formula:       $\text{Mean} = \frac{\text{sum of a set of items}}{\text{total number of items}}$

• Median:

  • Procedure: Arrange the numbers in numerical order. The median is the middle number.

  • Handling Even Sets: For an even number of values, the median is calculated as the mean of the two middle numbers.

• Mode:

  • Definition: The number in the data set that occurs most frequently.

Calculations of Central Tendency (Example 2)

Data Set #1

• Raw Data: $9, 12, 15, 7, 13, 12, 14, 12, 10, 8, 17$

• Order of Operations:

  • Sum = $9 + 12 + 15 + 7 + 13 + 12 + 14 + 12 + 10 + 8 + 17 = 129$

  • Total Items ($n$) = $11$

  • Mean Calculation: $\frac{129}{11} \approx 11.73$

• Numerical Order for Median: $7, 8, 9, 10, 12, 12, 12, 13, 14, 15, 17$

• Median: $12$

• Mode: $12$

Data Set #2

• Raw Data: $10, 12, 14, 8, 11, 15, 10, 12, 9, 13, 7, 10$

• Order of Operations:

  • Sum = $10 + 12 + 14 + 8 + 11 + 15 + 10 + 12 + 9 + 13 + 7 + 10 = 136$

  • Total Items ($n$) = $12$

  • Mean Calculation: $\frac{136}{12} \approx 11.33$

• Numerical Order for Median: $7, 8, 9, 10, 10, 10, 11, 12, 12, 13, 14, 15$

• Median Strategy for Even Set: Determine the mean of the $6\text{th}$ ($10$) and $7\text{th}$ ($11$) values.

  • Median Calculation: $\frac{10 + 11}{2} = 10.5$

• Mode: $10$

Measures of Spread

• Definition: Measures of spread summarize data by showing how scattered the values are and the extent to which they differ from the mean value.

• Key measures include:

  • Range.

  • Standard deviation.

  • Quartiles.

  • Percentiles.

• Application: Measures of spread are used in conjunction with measures of central tendency to provide a comprehensive understanding of the data.

Quartiles and Boxplots

• Quartiles:

  • Function: They divide a data set into four equal quarters, similar to how a median divides a set into halves.

  • Requirement: Data must be placed in numerical order from smallest to largest before quartiles can be identified.

• Boxplots:

  • Used to visually graph quartiles and the range of a data set.

• Spread Formulas:

  • Range: The difference between the maximum and minimum values.       $\text{Range} = \text{Maximum} - \text{Minimum}$

  • Interquartile Range (IQR): The difference between the third quartile ($Q_3$) and the first quartile ($Q_1$).       $\text{IQR} = Q_3 - Q_1$

The Five-Number Summary and Boxplot Steps (Example 3)

Execution Steps

1. Step 1: Order the data from smallest to largest.

2. Step 2: Find the median ($Q_2$), which splits the data into two halves.

3. Step 3: Find the median of the lower half ($Q_1$) and the median of the upper half ($Q_3$) to split the data into four parts.

4. Step 4: Identify the minimum and maximum values.

5. Step 5: Consolidate the five-number summary.

Data Set #1 Analysis

• Sorted Data: $7, 8, 9, 10, 12, 12, 12, 13, 14, 15, 17$

• Five-Number Summary:

  • Minimum (Lower Extreme): $7$

  • Lower Quartile ($Q_1$): $9$

  • Median ($Q_2$): $12$

  • Upper Quartile ($Q_3$): $14$

  • Maximum Value (Upper Extreme): $17$

• Calculated Metrics:

  • Range: $17 - 7 = 10$

  • Interquartile Range ($Q_3 - Q_1$): $14 - 9 = 5$

Data Set #2 Analysis

• Raw Data: $43, 82, 56, 78, 60, 57, 61, 83, 90, 66, 73, 60, 65, 68, 74, 80, 65, 71, 78, 82, 95$

• Smallest to Largest Sorting:   $43, 56, 57, 60, 60, 61, 65, 65, 66, 68, 71, 73, 74, 78, 78, 80, 82, 82, 83, 90, 95$

• Five-Number Summary ($n = 21$):

  • Minimum (Lower Extreme): $43$

  • Lower Quartile ($Q_1$): $60.5$ (Mean of $5\text{th}$ and $6\text{th}$ items)

  • Median ($Q_2$): $71$ ($11\text{th}$ item)

  • Upper Quartile ($Q_3$): $81$ (Mean of $16\text{th}$ and $17\text{th}$ items)

  • Maximum Value (Upper Extreme): $95$

• Calculated Metrics:

  • Range: $95 - 43 = 52$

  • Interquartile Range: $81 - 60.5 = 20.5$