Explanatory and Response Variables, Correlation (2.1)

Summary

This video script explores the concepts of explanatory and response variables, as well as correlation, in the context of statistical analysis. It explains how to use time plots and scatter plots to visualize the relationship between two variables, emphasizing the importance of identifying which variable is explanatory and which is responsive. The script delves into how correlation, denoted as 'R', measures both the direction and strength of a linear relationship between variables, and provides a step-by-step guide on calculating it using a formula. It concludes with a cautionary note on the potential for visual deception when interpreting correlation from scatter plots alone.

Takeaways

• 📊 Descriptive statistics like histograms, stem plots, and box plots are used to describe a single variable, while time plots and scatter plots are used to show relationships between two variables.
• 🌳 In a time plot, one variable is the explanatory variable (e.g., age of a tree), and the other is the response variable (e.g., tree height), indicating a cause-and-effect relationship.
• 📈 A scatter plot represents the values of two quantitative variables from the same population, with the explanatory variable typically on the x-axis and the response variable on the y-axis.
• 🔄 The terms 'explanatory variable' and 'response variable' can be used interchangeably with 'independent variable' and 'dependent variable', respectively.
• ⚠️ Not all data sets have an explanatory and response variable; some variables are unrelated and do not have a cause-and-effect relationship.
• 🔍 Correlation, denoted as R, measures the direction and strength of a linear relationship between two quantitative variables, independent of whether they are explanatory or response variables.
• ⬆️ A positive correlation indicates an upward slope in the data set, while a negative correlation indicates a downward slope.
• 🔢 The value of R ranges from -1 to 1, with -1 indicating a perfect negative correlation, 1 indicating a perfect positive correlation, and 0 indicating no correlation.
• 📝 The formula for calculating correlation involves the means, standard deviations, and the sum of the products of the differences from the means of the paired variables.
• 📚 To calculate correlation, gather data, create a table, calculate means and standard deviations, and apply the formula to find the correlation coefficient.
• 👀 Visual inspection of a scatter plot can be misleading; the actual value of R should be calculated and interpreted numerically rather than relying on visual assessment alone.

Q & A

What is the purpose of using histograms, stem plots, and box plots?

Histograms, stem plots, and box plots are used to describe one variable. They help in understanding the distribution and characteristics of a single dataset.

How can we compare two different populations with respect to the same variable?

We can use back-to-back stem plots and side-by-side box plots to compare two different populations with respect to the same variable, which helps in visualizing the differences and similarities between the two groups.

What is the difference between a response variable and an explanatory variable?

A response variable measures the outcome of a study, while an explanatory variable explains the outcome. For example, in a study about trees, the height of the tree is the response variable, and the age of the tree is the explanatory variable.

Why is a time plot used to show the relationship between two variables?

A time plot is used to show the relationship between two variables when there is a temporal relationship, where one variable changes in response to the other over time.

What is a scatter plot and how is it different from a time plot?

A scatter plot is a graphical representation of two quantitative variables from the same population, where each dot represents an individual's values for both variables. Unlike a time plot, a scatter plot does not necessarily have time on the x-axis.

Why is the explanatory variable usually plotted on the x-axis and the response variable on the y-axis?

The explanatory variable is plotted on the x-axis and the response variable on the y-axis because it represents the independent and dependent variables, respectively, with the explanatory variable influencing the response variable.

What is the role of correlation in data analysis?

Correlation, denoted as R, measures the direction and strength of a linear relationship between two quantitative variables. It helps in understanding how two variables move in relation to each other.

How is the direction of a data set's slope related to the value of R in correlation?

If a data set has an upward slope, R is positive, indicating a positive correlation. If it has a downward slope, R is negative, indicating a negative correlation. A perfect straight line slope results in R being either +1 or -1, representing perfect positive or negative correlation.

What does the value of R equal to 0 indicate in terms of correlation?

When R is equal to 0, it indicates that there is no correlation, meaning there is no linear relationship between the two variables.

How can we calculate the correlation between two variables?

Correlation can be calculated using a formula that involves the means, standard deviations, and the sum of the products of the differences from the means for each variable. The formula is more complex than it appears but follows a systematic process of calculation.

Why is it misleading to interpret correlation based solely on the visual appearance of a scatter plot?

Interpreting correlation based on the visual appearance of a scatter plot can be misleading because different scales can make the relationship appear stronger or weaker. The actual numerical value of R is needed to accurately determine the strength and direction of the correlation.

Outlines

00:00

📊 Exploring Variables and Correlation

This paragraph introduces the concepts of explanatory and response variables, as well as the idea of correlation. It explains how histograms, stem plots, and box plots can be used to describe a single variable, and how time plots and scatter plots can be utilized to examine the relationship between two variables. The explanatory variable is described as the one that influences the outcome, while the response variable is the outcome itself. The paragraph also clarifies that correlation, denoted by R, can be determined without strictly identifying explanatory and response variables. It discusses how correlation measures the direction and strength of a linear relationship between two quantitative variables, with positive and negative values indicating the slope of the relationship and values close to 1 or -1 indicating a strong linear relationship. The formula for calculating correlation is introduced, and an example of calculating it using a teacher's data on study hours and test scores is provided.

05:00

📈 Calculating and Interpreting Correlation

The second paragraph delves into the process of calculating the correlation coefficient, R, using a formula that involves the means and standard deviations of the variables in question. It provides a step-by-step guide on how to create a table for calculations, starting with finding the means of the X and Y values, then subtracting these means from each respective value to create deviations. These deviations are multiplied together and summed up to form part of the correlation formula. The paragraph emphasizes the importance of calculating standard deviations and correctly applying them in the formula to find the value of R. An example calculation is presented, resulting in an R value of 0.602, which indicates a moderate positive correlation. The paragraph also cautions against relying solely on visual interpretation of scatter plots to determine correlation, as different scales can affect perception, and stresses the importance of using calculated values for accurate interpretation.

Keywords

💡Explanatory Variable

An explanatory variable is a factor that is believed to influence another variable in a study. In the context of the video, the age of a tree is an explanatory variable because it is thought to explain the height of the tree. The video emphasizes that as the explanatory variable (age) increases, the response variable (height) also increases, illustrating a direct relationship.

💡Response Variable

A response variable is the outcome of a study that is measured to see if it is affected by the explanatory variable. In the video, the height of the tree is the response variable, which is measured to determine how it responds to changes in the explanatory variable (age). The video script uses this concept to explain how one variable can be the result of another.

💡Correlation

Correlation is a measure that expresses the extent to which two variables are linearly related. The video script explains that correlation, denoted as 'R', indicates the direction and strength of the relationship between two quantitative variables. It is a key concept in the video, as it is used to discuss the relationship between studying hours and test scores, and how it can be visually represented through scatter plots.

💡Scatter Plot

A scatter plot is a type of plot that shows the values of two variables for a set of data. Each dot on the scatter plot represents the values of the two variables for an individual data point. In the video, a scatter plot is used to illustrate the relationship between the number of hours studied and test scores, showing how each variable relates to the other without the constraint of time being on the x-axis.

💡Time Plot

A time plot is a specific type of graph where one of the variables is time, and it is used to show the relationship between two variables when one is believed to influence the other over time. The video script mentions time plots as a way to represent the relationship between the age of a tree (explanatory variable) and its height (response variable) over time.

💡Histogram

A histogram is a graphical representation of the distribution of a set of data, which the video script mentions as a way to describe one variable. It is used to show the frequency of data points within certain ranges or 'bins' and is a tool for understanding the distribution of a single variable.

💡Stem Plot

A stem plot is a graphical method to display data that consists of a 'stem' (a common part of all data points) and 'leaves' (the unique part of each data point). The video script refers to stem plots as a way to describe one variable and compares them to histograms and box plots for data representation.

💡Box Plot

A box plot is a standardized way of displaying the distribution of data based on a five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The video script discusses box plots as a tool for comparing two different populations with respect to the same variable.

💡Direction

In the context of correlation, direction refers to whether the relationship between two variables is positive (an upward slope) or negative (a downward slope). The video script explains that a positive correlation (R) indicates an upward slope, while a negative correlation indicates a downward slope, reflecting the nature of the linear relationship.

💡Strength

Strength in correlation refers to how closely the data points follow a straight line, indicating the degree of linear relationship between two variables. The video script clarifies that the strength of the relationship increases as the correlation coefficient 'R' approaches +1 or -1, with R = 0 indicating no linear relationship.

💡Formula

The formula mentioned in the video script is used to calculate the correlation coefficient 'R'. It involves several steps including calculating means, subtracting each value from the mean, multiplying the differences, and using standard deviations to arrive at the final 'R' value. The formula is central to the video's explanation of how to quantify the relationship between two variables.

Highlights

Explanatory and response variables are introduced as key concepts for understanding the relationship between two variables in statistical analysis.

Histograms, stem plots, and box plots are discussed as methods for describing a single variable.

Back-to-back stem plots and side-by-side box plots are used for comparing two populations with respect to the same variable.

Time plots are used to show the relationship between two variables when one is the outcome of the other.

The age of a tree is given as an example of an explanatory variable influencing the height, which is the response variable.

Scatter plots are introduced as a method to show the relationship between two quantitative variables without the need for time on the x-axis.

Each dot in a scatter plot represents an individual, illustrating the values of two variables for that individual.

The explanatory variable is typically plotted on the x-axis, and the response variable on the y-axis.

The concepts of independent and dependent variables are related to explanatory and response variables.

Examples are given where there is no need for explanatory or response variables, such as comparing unrelated events.

Correlation, denoted as R, is explained as a measure of the direction and strength of a linear relationship between two variables.

The direction of the slope in a data set determines whether the correlation is positive or negative.

Perfect positive and negative correlations are described with R values of +1 and -1, respectively.

The strength of the linear relationship is measured by how close R is to +1 or -1.

A formula for calculating the correlation coefficient is provided, emphasizing its importance in statistical analysis.

A practical example of calculating correlation between study hours and test scores is given, demonstrating the formula's application.

The importance of not relying solely on visual interpretation of scatter plots to determine correlation is stressed.

The concept that numbers do not lie is highlighted as crucial when interpreting correlation to avoid deception by graphs.

Transcripts

in this video we will be looking at

explanatory variables response variables

and

correlation in the first video set we

talked about how we can use histograms

stem plots and box plots to describe one

variable and we also used back-to-back

stem plots and side-by-side box plots to

help us compare two different

populations with respect to the same

variable these were good for describing

one variable but what about two

variables well we saw that we can use a

Time plot to show this if there is a

relationship between these two variables

one can be called the response variable

and the other can be called the

explanatory variable the response

variable measures the outcome of a study

and the explanatory variable explains

the outcome of a study in this example

the age of the tree is the explanatory

variable because as the tree gets older

the taller it will be so we can say that

the age of the tree EXP explains its

height this also means that the height

of the tree is the response

variable another way to show the

relationship between two variables is by

using a scatter plot unlike a Time plot

a scatter plot does not need to show

time on the x-axis a scatter plot shows

the values of two quantitative variables

that were measured from the same

population of

individuals this is a typical scatter

plot you can think of each dot as being

an individual so if we looked at this

individual we can see that this person

studied for 14 hours and got a test

score of 62 and this individual studied

for almost 6 hours and got a test score

of 30 you might have noticed that the

explanatory variable is always plotted

on the x-axis and the response variable

is always plotted on the y- AIS for this

reason the explanatory variable is

denoted as X and the response variable

is denoted as y you can also think of

the explanatory variable as being the

independent variable and the response

variable being the dependent

variable note that it is possible to not

have an explanatory variable and a

response variable for example the number

of points scored during a football game

versus the number of points scored

during a basketball game these are two

unrelated events and one variable does

not precisely explain the other if there

are no explanatory or response variables

then it doesn't matter where you plot

each variable on the graph this is

commonly seen when trying to compare two

unrelated variables or

events before we talk about correlation

I'd like to point out that when

determining correlation explanatory and

response variables are not necessary now

correlation is denoted as R and it tells

you about the direction and strength of

a linear relationship shared between two

quantitative variables correlation can

be expressed using Scatter Plots so

let's talk about how Direction and

strength is measured by correlation

we'll talk about Direction first

correlation tells us about the direction

or slope of a set of data so it tells us

if a data set has an upward slope or a

downward slope if we have an upward

slope we can say that R is positive if

we have a downward slope then R is

negative if we have an upward slope and

the data points follow a perfect

straight line then R is equal to pos1

and this is called a perfect positive

correlation in contrast if we have a

downward slope and the data points

follow a perfect straight line then R is

equal to -1 and this is called a perfect

negative correlation in both of these

cases we have a perfect linear

relationship correlation measures the

strength of this linear relationship we

can actually notice a pattern about how

correlation measures strength we saw

that R has values between positive 1 and

Nega one and we saw that the strength of

the linear relationship increased as R

got close to positive 1 or negative 1 so

when R is equal to Z this means that

there is no correlation in other words

there is no linear relationship

whatsoever we can see that as R gets

closer and closer to positive one the

linear relationship gets stronger and as

R gets closer and closer to -1 the

linear relationship also gets

stronger so how do you calculate

correlation correlation can be

calculated using this formula it seems

like a complicated formula formula but

it's easier than it looks so suppose a

teacher wants to determine the

correlation between the number of hours

spent studying and test scores to do

this he would first have to gather some

data I will assign the number of hours

spent studying as X and I will assign

the test scores to be y remember that

correlation doesn't care about

explanatory or response variables so it

didn't matter how I assigned these

variables because I would end up with

the same value of R when determining

correlation it's a good idea to make a

table to help you with your calculations

this table corresponds to the formula

specifically this part of the

formula so the first step is to

calculate the means for the X values and

the Y values which you should already

know how to do then we will subtract

each x value from xar so we will have 13

-

12.5 which is equal to

0.5 for the second row we will have 15 -

12.5 which is equal to

2.5 we will do this for each x value for

the Y values we would have 53 minus 68

which is equal to -15 we are basically

doing the same process for each yval for

the next step we will have to multiply

each of these terms together so we will

have 0.5 * -15 and for the second row we

will have 2.5 * 1 and so

on the next step is to add up all all

these products together and that gives

us

821 we can now plug this value into the

formula and since we added six products

and will be equal to six next we need to

calculate the standard deviations for

each variable and you should already

know how to do

this at this point we have all of the

ingredients we need for the formula so

we can plug in SX and Sy Y into the

formula and when we simplify this we get

our value of R which is equal to 0.602

so if we plotted our data it would look

like this we determined that R is equal

to

pos6 and this makes sense because each

data value seems to follow an upward

Direction be careful when trying to

interpret correlation by just looking at

the scatter plot when comparing these

graphs we might think that the scatter

plot on the right has a higher r value

because the data points are closer

together and it looks like it visually

displays a stronger linear

relationship unless R is equal to plus

or minus one it's really hard to

determine the value of R just by using

our eyes both of these graphs are

actually the same and they have the same

value of R they were just made using

different scales and this is why graphs

can deceive us using the notion that

numbers do not lie is applicable when

trying to interpret correlation