Unit 1 - Intro and Types of Variables

Categorical Variable

A categorical variable places things into groups or categories. The values are usually names or labels, not numbers that you would add or average.

They can be either nominal or ordinal.

This involves describing distributions with graphs (e.g. bar charts, pie charts).

• For example marital status, eye colour, favourite pet, province.

Numerical Variable (Quantitative)

A numerical variable contains numbers that represent a quantity or amount. You can do math with these values. Can be count or obtained by measurement, and it makes sense to perform arithmetic calculations, such as taking the average, on those values.

• This includes continuous variables, and discrete variables.

This involves describing distributions through using numbers. The variable will take a certain numerical value, and can be described in measurements (or number of units). This can be shown through histograms and time plots.

• Make sure does it make sense to perform arithmetic calculations on the average. Even if you are using number based things (e.g. birth month - categorical variable), it may not be relevant to do so (if the data is recorded in numbers).

  • If it is not numerical, then for sure you know that it is not quantitative.

• Examples include income level, age, height, test score, number of siblings.

Ordinal Data (One Type of Categorical Variable)

Data follows a natural order and the order makes sense. Categories that can be put in order and ranked. Is there a logical ordering that you should put one value in front of the other.

Examples:

• Education level: High school, college, university

• Satisfaction survey: Very Dissatisfied, Dissatisfied, Neutral, Satisfied, Very Satisfied

• T-shirt size: Small, Medium, Large

• Birth month: Jan, Feb, Mar, April, etc.

Nominal Data (One Type of Categorical Variable)

Data is recorded as labels. Categories that have no particular order or ranking. There is no category that is "higher" or "better" than another.

Examples:

• Eye , Brown, Green

• Province: Manitoba, Ontario, Alberta

• Type of pet: Dog, Cat, Fish

• Gender: Female, Male, Non-Binary

• Marital status: Single, Married, Widowed, Divorced

Population

The total amount of individuals or units about which we want information. The population is the entire group that you want information about. The whole group being studied.

Example:

• All students at your university.

• All residents of Manitoba.

• All children involved with a particular program.

Individuals (or Units)

Individuals are the people, animals, objects, or things being studied.

Example: If you are studying students in a classroom:

• Each student is an individual.

If you are studying cars in a parking lot:

• Each car is an individual

Variables

A variable is a characteristic or piece of information collected about each individual or unit. Variables can vary from individual to individual, and you can get different values.

Example: Suppose the individuals are students.

Variables could include:

• Age

• Height

• Eye color

• Grade level

• Location

What information are we collecting about each individual?

Sample

A sample is a smaller group taken from a population.

Example:

• Population: All 10,000 students at a university.

• Sample: 200 students selected from that university to complete a survey.

Researchers often use a sample because it is easier and less expensive than studying everyone in the population.

One Value of a Variable

A value is the specific information recorded for an individual.

For example, if the variable is Age:

Student	Age
Sarah	20
James	22
Emma	19

• Age is the variable.

• 20 is one value of the variable.

• 22 is another value.

• 19 is another value.

Another example:

• Variable: Eye Color

• Values: Brown, Blue, Green

Descriptive Statistics

Methods used to summarize and describe data so it is easier to understand. Instead of looking at every single piece of data, descriptive statistics give you a quick summary.

Example

Suppose five students received these test scores:

70, 75, 80, 85, 90

Descriptive statistics can tell us:

• Mean (average): 80

• Minimum score: 70

• Maximum score: 90

• Range: 20 (90 − 70)

These numbers help describe the data.

Questions to Ask (Categorical and Numerical Data)

1. If you were to have a numerical value, it does not tell you it is quantitative. But if is not numerical, then for sure you know it is not quantitative.

2. Is there a logical ordering between variables?

   • Example: Where someone is studying

Distribution

How frequently different values occur.

The way data values are spread out or distributed. The distribution of a v variable tells us what values it takes and how often it takes on these values.

It shows:

• What values occur

• How often they occur

• Whether values are clustered together or spread apart

  • This can be useful on histograms for using 5 - 10 intervals for datapoints.

Pie Chart

Visual representation of the relative frequency or proportion of the observed values. Proportions are values between 0 and 1. They can convert to percentages when multiple by 100.

• A proportion of the whole.

• Divide the specific number by the total number.

• Proportions (a number between 0 - 1) and can also be called relative frequency.

Bar Graph

A graph that uses bars to show and compare the frequencies (counts) of different categories.

Histogram

A histogram is a graph that shows how numerical data are grouped together. Asking how many values fall into certain ranges. The goal is to look at the overall distribution of the data (overall pattern).

Displays the distribution of values of a quantitative value. They display the counts of the number of data values that fall into specific “classes” or “intervals.”

For example, if you have students' ages:

18, 19, 19, 20, 20, 21, 22, 22, 23, 24

A histogram helps you see where most of the ages are concentrated.

What are intervals

Intervals (also called classes or bins) are groups of values. Instead of showing every age separately, we group ages into ranges. Every interval needs to be of equal length (e.g. using 10 every time).

• To decide on intervals you want to look at the smallest amount of data, and the largest amount of data, and pick break points in between the smallest and largest. You want to end up with 5 - 10 intervals.

For example:

Age Interval	Number of Students
18-19	3
20-21	3
22-23	3
24-25	1

Here:

• 18-19 is an interval.

• 20-21 is an interval.

• 22-23 is an interval.

The histogram would have a bar for each interval, and the height of the bar shows how many students are in that range.

Where to Place Exact Intervals (e.g. range is 10, and you have data that ranges from 30 - 40, but a number listed as 40)

If intervals fall range from 10, and you have data that falls into 30 - 40, it actually means that the interval is from 30 - 39.999. This means that 40 would fall into the next range of 40 - 50.

• Includes the left endpoint, but not the left endpoint.

Continuous Variable (Quantitative Variables)

**Most of the course will focus on this type.

Can take any value within a given range. Can take a percentage out of 100.

• Histogram is only used for this type of variable.

• This can include weight, height, test scores, distance.

Discrete Variable (Quantitative Variables)

Can only take a countable whole number of values.

• Examples include: Number of children in a family, number of days of rain in the month.

Proportions (Relative Frequency)

Proportion and relative frequency mean essentially the same thing. They tell you what part of the total a category represents.

Proportion = Frequency divided / by the total frequency

Suppose 20 students were surveyed about their favorite pet (10 + 6 + 4 = 20).

Pet	Frequency
Dog	10
Cat	6
Fish	4

Total = 20 students

For dogs: 10 divided by 20 = 0.50

• Frequency = 10

• Proportion (Relative Frequency) = 0.50

This means 50% of students chose dogs.

Converting to a Percentage

Multiply the proportion by 100:

0.50 × 100 = 50%

So:

• Proportion = 0.50

• Relative frequency = 0.50

• Percentage = 50%

What is the Difference Between Frequency and Relative Frequency?

Frequency is the actual count of how many times something occurs. The number of times a value appears.

Favorite Pet Survey

Pet	Frequency
Dog	5
Cat	3
Fish	2

• Frequency of Dog = 5

• Frequency of Cat = 3

• Frequency of Fish = 2

Relative frequency is the proportion or percentage of the total. The fraction or percentage of the total.

The formula is:

• Relative Frequency = Frequency divided / the total # of observations

Using the same example:

Total students = 5 + 3 + 2 = 10

Pet	Frequency	Relative Frequency
Dog	5	5/10 = 0.50 = 50%
Cat	3	3/10 = 0.30 = 30%
Fish	2	2/10 = 0.20 = 20%

To Make a Histogram

1. Look at the intervals and frequency

2. Show the data values between the decided interval (e.g. 2 between 30 - 40).

3. General format of histogram:

   1. Variables: On the x-axis (side).

   2. Count (Frequency): On the y-axis (bottom).

Three Elements that We Look at When Analyzing a Distribution

There are three main things we look at in analyzing a distribution:

1. Centre: Mean and Median

2. Shape: Approximately Symmetric, Skewed to the Left, Skewed to the Right

3. Spread: Measures of spread

Proportion Formula (Histogram)

What percentage of test scores were less than 60?

Proportion = # of scores that are less than 60 / total # of test scores

• Scores less than 60 = 10 test scores

• Total test scores = 40 test scores

• Divide 10/40 = 0.25 × 100 = 25%

Purpose of a Histogram (and Parts)

The goal is to not look at individual scores, but look at the overall pattern of the data. This includes looking at overall distribution of the data. We are looking at the centre of the distribution (can be looked at in several ways), and any deviation from this pattern.

This includes looking at the:

• Shape: This can include approximately symmetric, skewed to the left (highest on the right), skewed to the right (highest on the left)

• Any gaps: Intervals that does not have any values (no bar)

• Peaks: The highest point of the data

• Spread (of the data): From which value to which value does the data include. From the smallest value to the largest value of the dataset (e.g. the spread of data is from 30 - 100 because this includes the total amount of data that there is).

• Outliers: Observations that fall away from the overall pattern.

3 Different Distribution Shapes (Histogram)

A histogram can take a different shape depending on the data. This can include:

1. Approximately Symmetric: This happens when there is an approximate mirror image on either side of the highest peak.

   • Mean and the median are equal.

2. Skewed to the Left: When the distribution extends much farther out toward smaller data values. Lower values on the left, and higher values on the right. Peak on the right side (opposite).

   • Mean is less than the median. Observations on the tail are outliers. Outliers are pulling the mean in the certain direction.

3. Skewed to the Right: When the distribution extends much farther out toward larger data values. Larger values on the left side, as you go to the right the lower values appear. Peak on the left side (opposite).

   • Outliers are pulling the mean to the left. The mean ends up being larger than the median.

Timeplots

Gathering data in a sequence over a period of time, then you have a timeplot. This is looking at a trend, or an overall pattern. We look for trends, or any cyclical behaviours.

• Trends: Upward or downward between time periods (overall pattern).

• Cyclical Behaviours: Same pattern occurring over 12 months time point (every year the pattern is the same).

  • Seasonal variation is another term used for this (every 12 months seeing the patterns repeating themselves).

Features:

• X-axis = time (days, months, years, etc.)

• Y-axis = the variable being measured

• Points are connected with lines over time

How are Timeplots Different from Histograms?

Histogram: Shows how data are distributed. It answers the question How often do values occur?

• X-axis = numerical values or intervals

• Y-axis = frequency (count)

• Shows the distribution of the data

Timeplots: Shows how something changes over time. It answers the question How does a variable change as time passes?

• X-axis = time (days, months, years, etc.)

• Y-axis = the variable being measured

• Points are connected with lines

Central Tendency (Two Types)

The location of the centre of our data. This includes:

• Mean: The average value. Adding up all the data values, and dividing the sum by the data points. x-bar (x with bar on top). This can be considered as a balance point. xˉ("x-bar")

  • Not resistant to outliers, impacted when data values change.

• Median: The middle value. This is the value that has ½ the ordered data values as large or larger than it. And ½ the data the values as small or smaller than it.

  • Median is resistant to outliers. Not impacted when data values change.

  • notation = M

Remembering Where the Axis Is

"Y to the Sky"

The Y-axis goes up and down.

• "Y" and "Sky" rhyme.

• Y-axis = Sky-axis ☁

When you're looking at a graph, the line going up toward the sky is the Y-axis.

Deviations

Deviations are simply the distances between a data value and the average (mean).

Deviation = Data Value - Mean

• Deviations all have to add up to 0 by definition. The deviations always add up to 0 because the mean (average) is the balancing point of the data.

  • The values below the average create negative deviations.

  • The values above the average create positive deviations.

  • The mean is located exactly where these positives and negatives balance each other out.

• When we say that the seesaw would balance at 6, when we add up all of the deviations we get 0.

Example: Data = 10, 15, 20

Mean:

10 + 15 + 20 / 3 = 15

Now find the deviations:

Value	Deviation
10	10 − 15 = −5
15	15 − 15 = 0
20	20 − 15 = +5

Add the deviations: -5 + 0 + +5 = 0

Weighted Mean

How frequently we are observing a particular value.

In some cases when calculating the mean, some data values are given more weight than others.

• This can be because some values are observed more frequently, or because some values are is some sense more important than others (e.g. GPA).

• The notation is the wi. The weight is given to the i(th) of the value.

Median (& Steps)

The middle value in a set of ordered data. In ordered data the media is the value that splits the data into two equal parts. This is one of the other types of central tendency. M

Steps to finding the median

1. Order the data from smallest to largest.

2. Count the number of data values n and compute

   1. n + 1 / 2

3. Count the n + 1 / 2 data values from the smallest value to find the median.

n = the number of observations.

Median Steps

1. Order the data from smallest to largest

2. Count the # of data values n and compute n + 1 / 2

3. Count the n + 1 / 2 data values from the smallest value to find the median.

4. This step only occurs with an even number. This would involve adding the two numbers that something occurs in between:

   1. Add those two numbers together and divide by 2.

3 Types of Spread

Range: Full maximum - minimum = range. Covers the entire interval covering all 100% of data.

• Effected by outliers

Interquartile Range (IQR): Measure of spread or variability. It covers the middle 50% of the ordered data, so it is not affected by outliers.

• Not effected by outliers

Percentiles:

• Affected by outliers

Spread Ideas

Idea 1: Look at the lowest, highest values (range - R).

• This is the easiest method.

Idea 2: Highest / lowest may be outliers, so instead describe most of the data, and then also mention any outliers.

Range (Spread)

The range (R) is a measure of spread and is calculated from

maximum - minimum = final answer

• The larger the value of R, the more variable the data are.

• R measures the length of the interval containing all (100%) of the data.

• Range IS AFFECTED by outliers.

Finding the Median of a Quarters

1. Find Q3: Find the median = n + 1 / 2.

2. Look at all of the data values above 50% (half way) |, and find the median of that data.

For example: 3, 7, 9, 12 | 13, 14, 18, 26

12 + 13 = 25 / 2 = 12.5

The median is 12.5 with a range (R) of 26 - 3 = 23.

We can see that there are 4 numbers less than 12. 5, and 4 numbers greater than 12.5. Now what if we want to find the median of each of those groups of numbers?

Interquartile Range (IQR)

Measure of spread or variability. It covers the middle 50% of the ordered data, so it is not affected by outliers.

• Not affected by outliers because outliers tend to be a value very high or low, which would not impact the IQR.

• Q3 - Q1 = IQR

  • Q1: 25th percentile. 25% of all data values lie below it, and 75% of data lie above it.

  • Q3: 75th percentile. 75% of the data values lie below it, and 25% of data values lie about it.

Percentile (p th)

The percentile is the value such that the p% of data values lie below it and (100 - p)% of data values lie above it.

• Example 90th, means that 90% of values lie below it, and 10% of values lie above it (because it should equate to 100% in total).

• Example: Asking for the 60th percentile when n = 1,000.

  • (0.6)(1000) = 600

  • This method can only be used for the percentile, and NOT for the median.

How Do you Distinguish Between the Percentile, and the Score?

• Score = the actual value a person got. For example: A mark.

• Percentile = How that score compares to everyone else's scores. Your ranking compared to others.

Example: Suppose 100 students write a statistics test.

You score 80/100.

• Score = 80

• If your score is higher than 90 of the students, then you are in the 90th percentile.

  • Your score = 80

  • Your percentile = 90th percentile

• The percentile does not mean you got 90%.

• It means you scored better than about 90% of the students.

Five Number Summary

(M | FQ | M | TQ | M)

When describing distributions with numbers we can use the five number summary:

1. Minimum

2. First Quartile

3. Median

4. Third Quartile

5. Maximum

These numbers deserve the location (median), and the spread (range and IQR range).

THIS DOES NOT INCLUDE MEAN

Steps to Find the Five Number Summary

• Minimum = smallest value

• Q1 (First Quartile) = 25% of the data is below this value

• Median (Q2) = middle value

• Q3 (Third Quartile) = 75% of the data is below this value

• Maximum = largest value

This will look different depending on if it an odd, or even number observation.

Five Number Summary Example

Suppose the test scores are:

50, 60, 70, 80, 90, 100, 110

Step 1: Find the Minimum and Maximum

• Minimum = 50

• Maximum = 110

Step 2: Find the Median | n = 7 observations

n + 1 / 2

7 + 1 / 2 = 4 (count to the fourth number of the dataset to get the median within the data. This would be 80). This tells us that the median is located in the 4th position of the ordered dataset. The value in the 4th position is 80, so the median is 80. The middle value is not always 50. In this dataset, the middle value happens to be 80.

The median is the middle number.

50, 60, 70, 80, 90, 100, 110

The middle value is 80.

• Median = 80

Step 3: Find Q1

Look at the lower half of the data (do not include the median).

50, 60, 70

The middle of these values is 60.

• Q1 = 60

Step 4: Find Q3

Look at the upper half of the data.

90, 100, 110

The middle value is 100.

• Q3 = 100

Boxplot (Quantile Boxplot / Whisker Plot)

This is a was to summarize the distribution of data, showing the centre of the data, and spread and variability.

Consists of a rectangle that is formed by using the quartiles and whiskers extending from the rectangle to the maximum and minimum values.

• The whiskers that extend, encompass 100% of our data.

• Whiskers extend to the overall minimum and maximum.

Outlier Boxplot

1. Construct a box using the Q1, median and Q3.

2. Calculate the whiskers that extend to the upper and lower fence.

   1. Lower Fence (LF): Q1 - (1.5 x IQR)

   2. Upper Fence (UF): Q3 + (1.5 x IQR)

   3. Any value below Q1 - (1.5 X IQR), or above Q3 + (1.5 x IQR) is considered an outlier.

Standard Deviation

Written as s, this is the square root of the average squared deviation from the mean. Measure of spread around the mean. This means that you are adding up all the squared deviations.

Can be useful to summarize datasets that maybe require the mean as the measure of centre, we should also use SD as a measure of spread instead of range or IQR (which align with median).

• When the mean is being used for the dataset, then standard deviation should be used as spread.

  • Mean + Standard Deviation

  • Deviation: This is how far the data is from the mean.

• In your population your have variability and you want to estimate that variability within a larger population.

Sample Variance

This is written as s2, and it is the square of the standard deviation. Standard deviation tells you the typical distance from the mean. Variance tells you that same spread, but in squared units.

Variance measures how spread out the data is from the mean (average).

• Small variance = data values are close to the mean.

• Large variance = data values are far from the mean.