Variance in R

1.Variance

Finding the mean, median, and mode of a dataset is a good way to start getting an understanding of the general shape of your data

However, those three descriptive statistics only tell part of the story. Consider the two datasets below:

dataset_one <- c(-4, -2, 0, 2, 4)
dataset_two <- c(-400, -200, 0, 200, 400)
dataset_one

[1] -4 -2  0  2  4

dataset_two

[1] -400 -200    0  200  400

These two datasets have the same mean and median — both of those values happen to be 0. If we only reported these two statistics, we would not be communicating any meaninful difference between these two datasets.

This is where variance comes into play. Variance is a descriptive statistic that describes how spread out the points in a data set are.

Instructions

Run the code and take a look at the two histograms that get created. Also look at the mean of each dataset.

These two histograms show the test grades of students from two different teacher’s classes. While the datasets have very similar means, their variances are very different. Think about the following questions:

1.Which dataset looks the most spread out?

2.When looking at the spread of a histogram, why are the units on the x-axis so important?

3.What does the spread of the data tell you about these two teachers that you might not understand if you only looked at the mean?

# load libraries
library(readr)
library(dplyr)
library(ggplot2)

# load data
teacher_one_grades <- c(83.42, 88.04, 82.12, 85.02, 82.52, 87.47, 84.69, 85.18, 86.29, 85.53, 81.29, 82.54, 83.47, 83.91, 86.83, 88.5, 84.95, 83.79, 84.74, 84.03, 87.62, 81.15, 83.45, 80.24, 82.76, 83.98, 84.95, 83.37, 84.89, 87.29)
teacher_two_grades <- c(85.15, 95.64, 84.73, 71.46, 95.99, 81.61, 86.55, 79.81, 77.06, 92.86, 83.67, 73.63, 90.12, 80.64, 78.46, 76.86, 104.4, 88.53, 74.62, 91.27, 76.53, 94.37, 84.74, 81.84, 97.69, 70.77, 84.44, 88.06, 91.62, 65.82)

#plot data
t1_chart <- qplot(teacher_one_grades,
      geom='histogram',
      binwidth = .8,  
      main = 'Teacher One Grades', 
      ylab = 'Grades',
      fill=I("blue"), 
      col=I("red"), 
      alpha=I(.2)) +
  geom_vline(aes(xintercept=mean(teacher_one_grades),
                 color="mean"), linetype="solid",
             size=1) +
  scale_color_manual(name = "statistics", values = c(mean = "red" ))



t1_chart

t2_chart <- qplot(teacher_two_grades,
      geom='histogram',
      binwidth = .8,  
      main = 'Teacher Two Grades', 
      ylab = 'Grades',
      fill=I("blue"), 
      col=I("red"), 
      alpha=I(.2)) +
  geom_vline(aes(xintercept=mean(teacher_two_grades),
                 color="mean"), linetype="solid",
             size=1) +
  scale_color_manual(name = "statistics", values = c(mean = "red" ))



t2_chart

2.Distance From Mean

Now that you have learned the importance of describing the spread of a dataset, let’s figure out how to mathematically compute this number.

How would you attempt to capture the spread of the data in a single number?

Let’s start with our intuition — we want the variance of a dataset to be a large number if the data is spread out, and a small number if the data is close together.

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance.png")

A lot of people may initially consider using the range of the data. But that only considers two points in your entire dataset. Instead, we can include every point in our calculation by finding the difference between every data point and the mean.

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance1.png")

If the data is close together, then each data point will tend to be close to the mean, and the difference will be small. If the data is spread out, the difference between every data point and the mean will be larger.

Mathematically, we can write this comparison as:

difference = X − μ

Where X is a single data point and the Greek letter mu is the mean.

Instructions

1.We’ve given you a very small dataset of five values named grades. These are five randomly chosen grades from the first teacher’s class. We’ve also calculated the mean of this small dataset and stored it in a variable named mean.

Let’s find the difference between each of these data points and the mean. We’ve created a variable for each difference. Start with difference_one. Change the value of difference_one — it should be equal to the first value in the data set minus mean.

# load data
grades <- c(88, 82, 85, 84, 90)
mean <- mean(grades)

#Change these five variables
difference_one <- 88 - mean
difference_two <- 82 - mean
difference_three <- 85 - mean
difference_four <- 84 - mean
difference_five <- 90 - mean

print(paste("The mean of the data set is ", mean))

[1] "The mean of the data set is  85.8"

print(paste("The first student is", round(difference_one, digits=2) , "percentage points away from the mean."))

[1] "The first student is 2.2 percentage points away from the mean."

print(paste("The second student is ", round(difference_two, digits=2) , "percentage points away from the mean."))

[1] "The second student is  -3.8 percentage points away from the mean."

print(paste("The third student is",round(difference_three, digits=2) , "percentage points away from the mean."))

[1] "The third student is -0.8 percentage points away from the mean."

print(paste("The fourth student is",round(difference_four, digits=2) , "percentage points away from the mean."))

[1] "The fourth student is -1.8 percentage points away from the mean."

print(paste("The fifth student is",round(difference_five, digits=2) , "percentage points away from the mean."))

[1] "The fifth student is 4.2 percentage points away from the mean."

3.Average Distances

We now have five different values that describe how far away each point is from the mean. That seems to be a good start in describing the spread of the data. But the whole point of calculating variance was to get one number that describes the dataset. We don’t want to report five values — we want to combine those into one descriptive statistic.

To do this, we’ll take the average of those five numbers. By adding those numbers together and dividing by 5, we’ll end up with a single number that describes the average distance between our data points and the mean.

Note that we’re not quite done yet — our final answer is going to look a bit strange here. There’s a small problem that we’ll fix in the next exercise.

Instuctions

1.Sum the five variables difference_one through difference_five and store the result in the variable difference_sum.

grades <- c(88, 82, 85, 84, 90)
mean <- mean(grades)

mean

[1] 85.8

difference_one <- 88 - mean
difference_two <- 82 - mean
difference_three <- 85 - mean
difference_four <- 84 - mean
difference_five <- 90 - mean

#Part 1: Sum the differences
difference_sum <- difference_one + difference_two + difference_three + difference_four + difference_five

print(paste("The sum of the differences is ", format(difference_sum, scientific = FALSE )))

[1] "The sum of the differences is  0.00000000000001421085"

2.Divide difference_sum by 5 and store the result in the variable named average_difference.

Think about the answer. Do you think it accurately captures the spread of your data?

#Part 2: Average the differences
average_difference <- difference_sum / 5

#IGNORE CODE BELOW HERE
print(paste("The sum of the differences is ", format(difference_sum, scientific = FALSE )))

[1] "The sum of the differences is  0.00000000000001421085"

4.Square the Differences

We’re almost there! We have one small problem with our equation. Consider this very small dataset:

c(-5, 5)

The mean of this dataset is 0, so when we find the difference between each point and the mean we get -5 - 0 = -5 and 5 - 0 = 5.

When we take the average of -5 and 5 to get the variance, we get 0:

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance2.png")

Now think about what would happen if the dataset were c(-200, 200). We’d get the same result! That can’t possibly be right — the dataset with 200 is much more spread out than the dataset with 5, so the variance should be much larger!

The problem here is with negative numbers. Because one of our data points was 5 units below the mean and the other was 5 units above the mean, they canceled each other out!

When calculating variance, if a data point was above or below the mean — all we care about is how far away it was. To get rid of those pesky negative numbers, we’ll square the difference between each data point and the mean.

Our equation for finding the difference between a data point and the mean now looks like this:

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance3.png")

Instructions

1.Square each of the values stored in the variables difference_one through difference_five. In R, to square a number, use the ^ operator. The code below gives you an example of squaring the variable a.

squared_num <- a ^ 2

grades <- c(88, 82, 85, 84, 90)
mean <- mean(grades)

#Change these five variables
difference_one <- (88 - mean) ^ 2
difference_two <- (82 - mean) ^ 2
difference_three <- (85 - mean) ^ 2
difference_four <- (84 - mean) ^ 2
difference_five <- (90 - mean) ^ 2

#Part 1: Sum the differences
difference_sum <- difference_one + difference_two + difference_three + difference_four + difference_five

#Part 2: Average the differences
variance <- difference_sum / 5

#IGNORE CODE BELOW HERE
print(paste("The sum of the differences is ", format(difference_sum, scientific = FALSE )))

[1] "The sum of the differences is  40.8"

print(paste("The variance difference is " , format(variance, scientific = FALSE)))

[1] "The variance difference is  8.16"

5.Variance in R

Well done! You’ve calculated the variance of a data set. The full equation for the variance is as follows:

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance4.png")

Let’s dissect this equation a bit.

1.Variance is usually represented by the symbol sigma squared.

2.We start by taking every point in the dataset — from point number 1 to point number N — and finding the difference between that point and the mean.

3.Next, we square each difference to make all differences positive.

4.Finally, we average those squared differences by adding them together and dividing by N, the total number of points in the dataset.

All of this work can be done quickly using a function we provided. The variance() function takes a list of numbers as a parameter and returns the variance of that dataset.

variance <- function(x) mean((x-mean(x))^2)
dataset <- c(3, 5, -2, 49, 10)
var <- variance(dataset)
var

[1] 338.8

Instructions

1.We’ve imported the same two datasets from the beginning of the lesson. Run the code to see a histogram of the two datasets. This time, the histograms are plotted on the same graph to help visualize the difference in spread.

Which dataset do you expect to have a larger variance?

# Defining the variance of the population mean 
variance <- function(x) mean((x-mean(x))^2)

teacher_one_grades <- c(80.24, 81.15, 81.29, 82.12, 82.52, 82.54, 82.76, 83.37, 83.42, 83.45, 83.47, 83.79, 83.91, 83.98, 84.03, 84.69, 84.74, 84.89, 84.95, 84.95, 85.02, 85.18, 85.53, 86.29, 86.83, 87.29, 87.47, 87.62, 88.04, 88.5)
teacher_two_grades <- c(65.82, 70.77, 71.46, 73.63, 74.62, 76.53, 76.86, 77.06, 78.46, 79.81, 80.64, 81.61, 81.84, 83.67, 84.44, 84.73, 84.74, 85.15, 86.55, 88.06, 88.53, 90.12, 91.27, 91.62, 92.86, 94.37, 95.64, 95.99, 97.69, 104.4)

hist(teacher_one_grades, col=rgb(0,0,1,1/4),xlim=c(65,105), main="Teacher Grades One and Two", breaks=15)
hist(teacher_two_grades,  col=rgb(1,0,0,1/4), add=T, breaks=15)

legend("topright", c("Teacher 1", "Teacher 2"), fill=c("blue", "red"))
box()

2.Scroll down in the code to find where we’ve definied teacher_one_variance and teacher_two_variance. Set those variables equal to the variance of each dataset using the variance() function.

# Set these two variables equal to the variance of each dataset using the var() function in R
teacher_one_variance <- variance(teacher_one_grades)
teacher_two_variance <- variance(teacher_two_grades)

print(paste("The mean of the test scores in teacher one's class is ", mean(teacher_one_grades)))

[1] "The mean of the test scores in teacher one's class is  84.4676666666667"

print(paste("The mean of the test scores in teacher two's class is ", mean(teacher_two_grades)))

[1] "The mean of the test scores in teacher two's class is  84.298"

print(paste("The variance of the test scores in teacher one's class is ", teacher_one_variance))

[1] "The variance of the test scores in teacher one's class is  4.26651788888889"

print(paste("The variance of the test scores in teacher two's class is ", teacher_two_variance))

[1] "The variance of the test scores in teacher two's class is  78.1319893333334"

6.Review

Great work! In this lesson you’ve learned about variance and how to calculate it.

In the example used in this lesson, the importance of variance was highlighted by showing data from test scores in classes taught by two different teachers. What story does variance tell? What conclusions can we draw from this statistic?

knitr::include_graphics("C:/Users/kuoan/Desktop/R Code/Variance5.png")

In the class with low variance, it seems like the teacher strives to make sure all students have a firm understanding of the subject, but nobody is exemplary.

In the class with high variance, the teacher might focus more of their attention on certain students. This might enable some students to ace their tests, but other students get left behind.

If we only looked at statistics like mean, median, and mode, these nuances in the data wouldn’t be represented.

Instructions

1.You might be wondering why we need to compute the variance. After all, by comparing histograms, it was fairly easy to tell which dataset had a larger spread.

Variance is useful because it is a measure of spread. While we might get a general understanding of the spread by looking at a histogram, computing the variance gives us a numerical value that helps us describe the level of confidence of our comparison.

It is also interesting to consider datasets that don’t have the same general curve.

Run the code to see two datasets that have a similar mean, but look very different. You’ll also see a printout of their mean and variance. Before looking at the printout, try to guess which dataset has a larger variance.

dataset_one <- c(-1.00271051e+01,  2.44056436e+00,  2.94492779e+00,  2.30809322e+01,
        1.44228252e+01, -6.62170391e+00, -5.84145412e+00, -5.82085113e+00,
       -6.23441951e+00,  4.89589152e+00,  5.53333473e+00,  8.13436224e-01,
        8.31733744e+00,  1.26817491e+01,  1.63690746e+01,  9.27975035e+00,
        3.38153801e-01, -2.08387710e+00, -3.41709471e+00, -1.16798232e+01,
       -1.27378389e+01, -1.04388072e+01,  3.04246862e+00, -1.95272242e+01,
        1.28730457e+01,  2.16461123e+01, -5.36036891e+00,  7.62597179e+00,
       -5.26893421e+00, -2.14136511e+00,  1.21484045e+00, -1.46287517e+01,
        5.55387760e+00, -1.27708511e+01, -8.77405168e+00,  3.03872494e+00,
        1.69469922e+01, -7.75303856e+00,  4.38781565e+00,  1.22995899e+00,
       -1.53206965e+01,  4.41561196e+00,  5.25609049e+00,  7.84277699e+00,
       -2.07648488e+01,  1.06480808e+01,  4.58888574e+00,  9.91680236e+00,
        1.89070643e+01, -2.19369193e+01, -1.68046317e+01,  7.98651903e+00,
        8.70712472e+00,  1.53906517e+01, -5.06645423e+00,  1.21229692e+01,
       -6.89022105e+00, -1.89235490e+00,  1.05736281e+01, -3.34405734e-01,
        9.18688220e+00, -5.28673711e+00, -6.96617094e+00, -2.93475354e+00,
       -4.11820660e+00,  6.42745765e+00,  2.35309576e+00, -6.18181195e+00,
        1.89856474e+01,  1.59507619e+01,  1.48997068e+01, -9.45567486e+00,
       -8.65923566e-01,  6.58553224e+00, -7.19112835e+00,  7.94435228e+00,
        7.80545934e+00,  1.32741447e+01, -6.36203259e+00, -5.97402732e-01,
        1.41425808e+01, -9.88711845e+00, -2.56723609e+01, -3.53122817e+00,
       -3.35992879e+00,  2.52457751e+01, -1.14729375e+01,  3.29799035e+00,
        3.61372888e+00, -1.34757379e+01,  1.57199355e+01,  1.16738057e+01,
       -8.46912595e+00, -9.94020992e-01, -7.62195108e+00, -1.58112435e+01,
       -6.11703783e+00,  1.91273643e+01, -2.97716951e+00, -1.59703952e+01,
       -2.47215710e+00,  2.68675569e-01,  2.51171550e+01, -4.76331016e+00,
       -1.46726833e+01,  1.40317129e+01, -3.82659938e-01,  3.18227889e+00,
       -1.76638891e+01, -2.23390103e+00,  8.03838894e-01,  6.67611722e+00,
       -1.64902139e+00,  5.16742707e+00, -1.52248250e+01,  1.21547485e+01,
       -7.63311657e+00,  1.09871453e+01, -1.37862013e+00,  2.55144651e+01,
       -1.51431679e+00, -7.30840019e+00,  2.14850396e+01, -2.38610237e+01,
        1.59614887e+01, -1.35501679e+01,  1.19346760e+01,  7.87909561e+00,
       -2.31891764e+00,  1.33466581e+01,  9.43822950e+00, -2.78843356e+00,
        1.06501154e+00,  9.62415715e+00, -1.42182422e+01, -1.20659874e+01,
       -4.46565612e-02, -1.25939923e+01, -9.58473020e+00,  1.53957060e+01,
       -9.52262657e+00,  2.76362127e+00, -1.71650925e-01, -1.73972407e+01,
       -2.79329394e+00,  6.56918883e+00,  1.89830025e+01, -6.77696679e+00,
        5.91484218e+00,  5.39165239e+00,  1.32198781e+01,  3.87114403e+00,
       -1.48986640e+01, -1.47566325e+01,  2.02663507e+01,  1.54087490e+01,
       -5.81149983e+00,  1.86805163e+01,  3.78225084e+00, -9.12508124e+00,
       -1.73796803e+01,  1.36587933e+01, -1.46087181e+01,  2.97667538e+00,
        1.11217091e+01,  7.20572710e+00,  5.55445139e+00,  4.33888556e+00,
        1.61314695e+01,  4.90812689e+00,  1.70667051e-01, -9.05775142e+00,
        1.04576939e+01,  1.17630929e+01, -1.14281811e+00,  8.09986292e+00,
        2.47784977e+01, -2.74775064e+00,  1.17933710e+01,  6.60244794e+00,
        6.60722780e+00,  2.95871213e+00,  1.61326239e+01, -1.38411266e+01,
       -4.90393228e-01,  1.45098476e+01, -9.63981447e-01, -6.15608177e+00,
       -1.48120973e+01, -1.03689828e+00,  1.08325190e+01, -1.76544252e+01,
        8.41791411e+00,  1.04969722e+01, -1.17403874e+01, -6.07773625e+00,
       -1.29702416e+00,  9.88515381e+00, -7.45412632e-01, -3.83020600e-01,
        9.50632014e-01,  1.41993910e+00,  5.31411990e+00,  1.14706188e+01,
        2.78739888e+00,  1.13591669e+01, -4.75973873e+00,  1.98834344e+01,
        5.20357813e-01, -1.20938541e+00, -1.13374036e+01, -8.73216588e+00,
       -4.78878436e+00, -3.14456466e+00, -4.68250992e+00, -1.19700829e+01,
        3.46368334e+00,  1.44383313e+01, -1.49386238e+01, -5.32684068e+00,
        1.37936684e+01,  3.97472819e+00,  1.00063184e+01, -1.81694929e+00,
        7.28905043e+00, -1.08610104e+00,  1.92432996e+00,  2.79125753e+00,
       -1.73890274e+01,  1.89147773e+01, -6.99051689e+00,  5.47632468e+00,
        6.09804762e+00, -4.21367676e+00, -1.00852313e+01,  7.05299358e+00,
        1.14465605e+01,  8.92852343e+00, -2.94236446e+00,  2.20109552e+01,
       -9.17153844e+00,  1.87965961e+00,  7.18825521e-01,  1.58438617e+01,
        1.79953895e+00, -6.76733029e+00,  1.64240833e+01, -9.89727591e+00,
       -1.92693865e+00, -1.01219428e+00, -2.67988793e+00,  5.42485293e+00,
        1.06884117e+01,  7.12514376e+00, -1.44709989e+01, -1.19428184e+01,
        4.82603084e+00,  7.59922180e+00,  4.97424484e+00, -2.86732307e+00,
       -6.11822826e+00,  8.00873616e+00,  1.25610588e+01, -4.80549112e+00,
       -5.76916012e-01,  1.03644120e+01, -1.43980492e+01, -3.15428498e+00,
        1.54574935e+01, -1.42344661e+01, -6.93905566e+00,  1.36849545e+01,
       -1.08938282e+01, -3.88565904e+00,  7.31887107e+00,  7.84277899e+00,
        3.01937016e+00, -7.83974066e+00,  3.91146829e+00,  1.11633983e+01,
       -1.08952105e+01,  6.54147200e+00,  7.70121590e+00, -1.06854052e+01,
       -1.90740705e+01, -4.43893087e+00,  6.72480217e+00,  6.58522624e+00,
        5.69611295e+00,  2.30096956e+00,  7.23617164e+00, -6.81065939e+00,
       -4.73135791e+00, -1.80566510e+00,  1.56202635e+01,  3.05420007e+00,
       -2.60134119e+00, -7.42230559e+00, -6.17447657e+00, -5.11683609e-01,
        1.03083521e+00,  1.24560826e+01,  3.31472666e+00,  1.48450236e+00,
        4.83288494e-02, -3.49263069e+00, -6.83129468e+00, -9.37877698e-01,
        4.33654667e+00,  6.79851131e+00,  2.01220489e+01,  6.50211319e+00,
       -8.87815308e+00, -7.00733331e+00, -3.63802320e+00, -5.89664106e-01,
        4.87791552e+00,  1.84580922e+01,  8.05774721e+00,  3.54965347e+00,
       -7.97175780e+00,  1.51516610e+01,  4.82352356e+00,  4.03725946e+00,
       -3.03939858e+00, -6.33957489e+00, -9.79801424e+00, -1.96423393e+01,
        1.77531979e+01,  3.29261158e+00, -6.66092286e+00,  2.62131813e+00,
       -3.08687746e+00, -5.64599116e+00, -8.60246717e-01, -1.34225530e+01,
        4.86575872e+00, -3.15902152e+00, -2.60504880e+01, -2.84565119e+00,
        2.85349082e+00,  1.29494578e+01, -4.03744657e+00,  3.65363430e+00,
       -7.17208453e+00,  1.56662828e+01,  1.10308018e+00, -1.27705137e+01,
        3.92076405e-01,  1.27482625e+01,  1.31549433e+01,  2.02271678e+01,
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       -1.64235530e+01,  3.50246546e+00, -5.83717126e+00, -1.14935559e+00,
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        5.81669161e+00,  5.66729707e+00,  6.09862160e+00, -7.39287428e+00,
        3.48798116e+00,  2.76832021e+00, -8.85766523e+00,  3.39080367e+01,
       -1.76083810e+01,  6.21020926e+00,  2.54031129e+01, -1.83956595e+00,
        8.53923687e+00, -4.92577266e+00,  9.76502352e+00, -6.74546638e+00,
       -9.10076756e+00, -5.86704591e+00,  1.10937027e+01,  4.20202631e+00,
       -9.07280275e+00, -1.89147761e+01, -2.13414108e+00, -1.31430323e+01,
       -8.54243350e+00, -7.15994637e+00,  1.72466804e+01,  1.99967542e+01,
        1.18470785e+01, -1.35280968e+01, -8.30031184e-02, -2.30994801e+01,
        6.84897426e+00,  7.56798128e+00,  1.10418412e+01, -1.07024141e+01,
        1.69611551e+01, -1.45867194e+01, -2.24250005e+00,  2.72361022e+00,
       -1.37618539e+01, -1.86866265e+00,  7.57366825e+00, -2.38119276e+00,
        1.34935923e+01, -6.32820839e+00, -1.30020319e+01, -4.00560595e+00,
       -1.06719029e+00,  5.02873493e-01, -8.37662132e+00, -1.86577516e+00,
        1.19251055e+01, -2.11653739e+01, -4.88404880e+00,  3.68131765e+00,
        5.66894443e+00,  5.31228636e+00, -4.58883020e+00, -1.69383376e+00,
        7.84212530e+00,  4.40700820e+00, -1.48488074e+01, -1.03504577e+01,
        1.49626024e+01,  1.10800825e+01, -1.38241677e+00, -5.59610848e-01,
        8.83213550e+00,  3.43572857e+00,  4.12853715e+00, -7.55509369e+00,
       -2.94806301e+00,  1.02204334e+01, -1.01593548e+01, -1.99263951e+01,
       -1.13819331e+01,  3.73323129e-01,  1.01285129e+01,  8.88002853e+00,
        1.01969430e+01,  7.94264570e-01,  1.30334624e+01, -5.70290959e+00,
        3.27483299e-01, -2.26551913e+00,  1.17664388e+01,  4.30850727e+00,
        1.00712603e+01,  3.49658728e+00,  4.47666092e+00, -1.56545292e+00,
        1.51232472e+00,  4.69460730e+00,  2.25242778e+01, -7.73318339e+00,
       -6.36161047e+00,  6.22469371e+00, -3.41633188e+00,  7.90870268e+00,
        3.45426444e+00,  5.52389659e+00, -5.15546064e+00,  1.27962722e+01,
        1.25838459e+00, -3.73253883e+00, -1.80130117e+00,  5.79008834e-01,
        9.48809366e+00,  1.59939226e+01, -1.47967929e+01, -1.48592818e+01,
       -2.99116188e+00, -9.20077268e+00, -1.37495896e+01, -4.05906754e+00,
        1.76055491e-01, -2.85789496e+01, -7.21077408e+00,  1.70543890e+01,
        5.27772388e+00, -7.25293134e+00, -7.37325351e+00,  1.05334124e+00,
       -1.78416362e+01,  9.28105828e+00,  1.26437070e+01,  1.54551188e+01,
       -1.10949217e+01,  1.79983371e+00,  5.38921688e+00, -9.63352540e+00,
        2.05339209e+00,  6.03188131e+00, -2.52931140e+00, -5.24876159e+00,
       -9.46397021e+00,  5.99528521e-02, -8.48974189e+00,  7.47513988e+00,
        1.41873164e+01,  1.41672752e+01,  6.66535383e+00, -7.52790260e+00,
       -2.15670119e+01,  1.51947630e+01,  2.23328414e-02, -3.14316541e+00,
       -2.11541305e+01, -1.12726062e+01,  1.44591448e+01, -1.12796072e+01,
       -8.19827984e+00, -6.04143972e+00,  6.15224257e+00,  1.61281571e+01,
        3.94540935e-01,  1.90810190e+01, -3.19162611e+00, -9.08011137e+00,
        1.07921551e+00, -9.09914573e-01, -8.00308495e+00,  1.11465749e+01,
        3.61608042e+00,  8.74967388e-01, -9.04283165e+00,  1.24063484e+01,
        7.99920126e+00, -1.13971484e+01,  7.53527435e+00, -1.46630406e+01,
       -1.79285585e+00,  7.36937658e-01,  1.09867806e+01, -1.32986971e+01,
       -7.31117358e+00,  9.58012106e+00,  7.60995491e+00,  1.45008626e+00,
       -9.59971205e+00, -1.93366148e+01, -2.85224418e+00, -1.01309003e+01,
       -1.28534491e+01,  1.05144042e+01, -1.48253688e+01,  1.41778979e+00,
        2.91551432e-01,  2.17542966e+01, -1.85807198e+01,  7.49897579e+00,
       -6.91107789e+00,  8.99179663e+00, -4.20610627e+00,  1.50957141e+00,
       -5.94399217e+00, -8.28086920e+00,  1.05354370e+01, -3.63356240e-02,
       -2.24759853e+00,  2.15892639e+01,  1.82974662e+00, -5.90103851e+00,
       -4.75040679e+00,  3.29300984e-01,  1.07398248e+00, -1.57457823e+01,
        1.05529738e+01,  2.67801666e+00, -1.80751781e-01,  8.12202814e+00,
       -6.75222101e+00, -2.40550303e+00,  5.22337301e+00,  1.04694094e+01,
        4.84373399e+00,  6.44931126e+00, -3.77783270e+00, -4.45269843e-01,
       -3.65605721e+00,  7.65182422e+00,  1.22449531e+01, -3.47946569e+00,
       -1.72170640e+00, -4.81774940e-01, -1.20966905e+01,  4.72043903e-01,
       -4.55877880e+00,  1.28387373e+00,  1.05033182e+01,  3.34580114e-01,
       -1.60785915e+00, -2.45263492e+01, -1.22825288e+01,  6.61804601e+00,
       -1.90130974e+00,  7.28087346e+00, -1.57384721e+00,  5.70132945e+00,
        2.19176464e+00,  7.95998779e+00,  6.40228719e+00, -8.81958887e+00,
        3.48367567e-01, -1.73685276e+01,  1.10102313e+01,  6.88460706e+00,
       -3.55209278e-01, -3.50037634e+00, -4.44480626e+00, -1.60970824e+01,
       -1.61281452e+01,  3.58321766e+00,  3.63209284e+00, -1.00896986e+01,
       -1.15960651e+01, -7.82079367e+00, -9.23927352e+00, -4.50970376e+00,
        3.08664319e+00, -4.30847194e+00, -1.21292184e+01, -3.36527091e+00,
       -1.36494720e+01, -1.82072851e+01,  8.23806397e-01,  6.17703959e+00,
       -1.38109673e+01, -1.02484845e+01, -3.47608353e+00,  1.46616797e+00,
        5.40173229e+00, -1.15148072e+01,  1.07270278e+01,  5.51330585e+00,
       -1.38885129e+00, -1.31119972e+01,  9.52341993e+00,  1.97869719e+00,
       -1.57895974e+01, -2.90513169e-01, -1.48225405e+01, -2.58355696e+00,
        2.93504846e+00,  1.03214208e+01, -5.31085373e+00,  2.30917392e+01,
       -1.56413322e+01,  6.81691363e+00, -2.85739714e+01, -5.34739093e+00,
        3.58691115e+00, -5.20053505e+00, -1.16591022e+01,  1.08674243e+01,
        1.44370122e+01,  7.65073545e+00, -7.95189061e+00,  6.92777757e+00,
        2.59120763e+00, -1.29875074e+01, -9.11002025e+00,  4.34387234e+00,
       -1.07659697e+00, -2.82874599e+00,  4.60262770e+00,  9.96668833e+00,
       -1.48422639e+01, -4.28449151e+00,  9.64930304e-02,  1.29337374e+01,
       -3.29967576e+00, -4.66947667e+00,  4.20788365e+00, -1.36044081e+01,
       -1.40948293e+01,  2.06740209e+01,  1.36978032e+01, -3.97761769e+00,
        1.48926453e+01,  2.36048250e+01,  4.74697175e+00, -1.58790599e+01,
        8.76697874e+00,  1.59804895e+01, -1.01043803e+00, -7.72542598e+00,
        5.42541698e+00,  1.79628672e+01,  1.12465736e+01,  2.12474864e+00,
       -2.51921790e+01,  6.16241434e+00, -5.07002240e+00, -1.32278974e+01,
       -9.73396828e+00,  9.00735181e+00,  2.28199871e+00, -2.25495705e+00,
       -2.34536207e+00, -9.95846355e+00, -5.98077096e+00, -1.06542997e+01,
        1.80918391e+00,  4.67861654e+00,  3.36477645e+00,  5.22855433e+00,
        9.20732424e+00, -5.74235818e+00,  4.52639479e+00,  1.47198710e+01,
        3.21596956e+00, -1.61500420e+01,  1.13630551e+01, -4.11270948e+00,
       -3.01963797e+01, -2.62679064e+00,  7.84996194e+00, -7.34528723e+00,
        7.43574926e+00,  3.80724070e+00, -1.41458712e+01,  7.06578310e+00,
       -7.25614001e+00,  9.72760518e+00, -7.56175774e-01, -9.44654557e+00,
        1.10230378e+00, -1.21545279e+01, -1.41149989e+01,  6.46651834e+00,
       -1.59871620e+00, -1.08007310e+01, -4.73618182e+00,  1.07336116e+00,
       -2.65878505e+00, -6.12868812e+00, -5.54011942e+00, -1.17907590e+01,
       -5.50607109e+00, -1.98937443e+01,  1.22444575e+01,  2.47374093e+01,
        4.75891560e+00, -1.18988639e+01, -2.64141146e+01,  1.00567234e+01,
        8.48931463e+00, -1.35494234e+01,  8.44139461e+00, -8.92792181e+00,
       -1.01477627e+01,  1.29144717e+01,  7.66801332e+00, -6.62807798e+00,
        2.71921135e+00, -1.60504294e-01,  5.49202042e+00,  7.13055822e+00,
       -1.72658521e+01,  3.71052801e+00,  1.07536203e+01,  9.30084874e+00,
       -1.91799142e+00, -4.74488093e-01,  2.86087865e+00, -1.24863568e+01,
       -7.63095780e+00, -2.55692028e+00, -1.85436182e+00,  7.62152486e+00,
        3.63014951e+00, -1.06408260e-01, -2.33375973e+00,  3.47213341e+00,
        5.24250134e+00, -2.53880279e+00,  3.41772141e+00,  1.93044683e+00,
        9.71030725e+00,  7.50400208e+00, -4.04647307e+00, -1.39492332e+01,
        1.49168911e+01,  4.01025381e+00, -2.65411984e+00,  6.32343056e+00,
       -2.99724524e+00,  7.13999712e+00, -1.35688224e+01, -1.43245460e+01,
       -7.18042915e+00, -6.80238018e+00,  4.23794481e+00,  9.56817039e+00,
        4.08089878e+00, -1.25890118e+01,  5.88379306e+00,  1.17982501e+01,
        1.39246820e+00,  9.65872575e+00, -1.09193220e+01,  6.45054920e+00,
       -7.82783581e+00, -3.21995205e+00, -1.11545861e+00,  2.05621100e+00,
       -6.82374161e+00,  7.60886659e+00,  1.10514008e+01,  2.18351847e+00,
        5.30625601e+00,  1.97192156e+00, -8.57635003e+00, -1.06824735e+01,
       -7.90165663e-01, -3.54504376e+00, -6.29044369e+00, -1.49051466e+01,
        8.86942900e+00,  1.06026146e+01, -2.58697560e+00, -3.31822291e+00,
        6.27672629e+00, -1.60308782e+01, -4.45769702e+00, -3.05045332e+00,
        1.31419478e+00,  1.53111852e+00, -5.87674832e+00,  1.58950138e+01,
        5.19551476e+00,  8.05734657e+00,  1.41070460e+01,  2.80779312e+00,
       -3.85052697e+00, -1.90682904e-01, -2.77435805e+00, -2.59678694e+01,
       -1.18195358e+01, -1.14589779e+01, -4.19070768e+00, -1.12700408e+01,
        1.56961957e+00, -2.54573875e+00, -5.18418281e+00,  2.82351842e-01,
       -1.03672944e+01,  8.58548766e+00, -2.94033676e+00,  9.42349290e-01,
        3.18940249e+00,  1.12918667e+01,  1.34528214e+01,  1.16628006e+01,
        4.51989443e+00,  1.38670589e+01,  2.26617926e+01, -9.42311801e-01,
       -2.97260135e+00,  7.19267537e+00, -7.44067031e+00,  1.20598929e+01,
        2.91843389e+00,  9.03552285e+00, -2.00439714e+01,  3.54853135e+00,
        1.32150785e+01,  1.53797967e+01, -4.70132880e+00,  1.32794919e+01,
        4.53960246e+00, -6.37357578e+00,  1.33646667e+01,  8.48142780e+00,
       -2.07205316e+00, -5.65068335e+00, -4.20980577e+00,  5.09511825e+00,
        2.82824641e+01, -1.29339951e+01, -4.14335656e+00, -8.65784726e+00,
       -2.14843615e+00, -8.29071513e+00,  1.41632832e+01, -7.10015513e+00,
        9.77624870e+00,  6.56837600e+00, -4.31541037e+00,  3.84246312e+00,
       -7.42758485e+00,  4.57557581e+00,  1.59253555e+00, -5.02641814e+00,
        1.86179078e+01, -8.06343548e+00,  3.54604742e+00,  5.75968564e+00,
       -1.04462240e+01,  5.95977993e+00,  5.90130964e+00, -1.06428557e+01,
        9.09042339e-01,  4.44019657e+00,  1.06078791e+01, -1.39302809e+01,
        1.12125155e+01,  5.76468955e+00, -6.53443096e+00, -1.90288547e+01,
       -2.02012499e+00,  2.62844642e+00,  4.30723455e+00, -6.01793746e+00,
        7.53845641e+00,  1.03150494e+01,  4.16377847e+00,  9.74113168e+00,
       -5.01529321e+00,  4.65107794e+00, -4.15315319e+00,  4.92241564e+00,
        2.03071442e+01,  5.23442195e+00, -3.29801625e+00, -7.42382230e+00,
       -1.70240624e+01, -4.34773172e+00,  6.34241073e+00, -4.07488513e+00,
        2.80985261e+00, -3.63249370e+00,  1.36274014e+00, -7.29153259e+00,
       -6.33076714e-01,  1.52835230e+01,  8.40499230e+00, -3.82934498e+00,
        6.38432428e+00, -3.91549431e+00, -1.42103664e+00,  2.24129059e+00,
       -5.39223713e-01,  8.90372071e+00,  2.79462795e+00,  7.30155029e+00,
       -8.13010370e+00,  1.10007284e+01,  2.85226698e+00,  1.92329063e+01,
        4.15862035e+00, -7.16868017e+00, -1.79475322e+01, -3.44962219e+00)

dataset_two <- c(-1.39544989e+01, -1.11497737e+01, -1.55207724e+01, -7.07941890e+00,
       -8.78488008e+00, -1.45445995e+01, -6.76869063e+00, -1.24266270e+01,
       -6.49969549e+00, -5.08372531e+00, -7.31388499e+00, -3.28617609e+00,
       -1.10499714e+01, -1.11495237e+01, -1.46785600e+00, -1.23642437e+01,
       -2.45948278e+01, -1.05550516e+01, -1.65632776e+01, -1.23707353e+01,
       -6.40467281e+00, -1.41890523e+01, -1.34541121e+01, -1.38638031e+01,
       -1.18830255e+01, -1.07134349e+01, -1.15641491e+01, -1.00523919e+01,
       -1.18902202e+01, -1.18603660e+01, -7.37306484e+00, -6.90372448e+00,
       -9.60599428e+00, -6.58705859e+00, -7.42990681e+00, -2.00724841e+01,
       -7.92165792e+00, -7.30027523e+00, -9.61253073e+00, -9.94926610e+00,
       -7.40318597e+00, -6.40128501e+00, -9.50255053e+00, -4.00046384e+00,
       -1.23193172e+01, -1.46969218e+01, -1.80865629e+01, -3.81875547e+00,
       -1.53203447e+01, -8.83697274e+00, -9.89478204e+00, -8.31986023e+00,
       -6.65839945e+00, -6.81534268e+00, -1.21999892e+01, -1.73570073e+01,
       -6.00909079e+00, -1.05052092e+01, -9.90103569e+00, -1.14222935e+01,
       -1.55690143e+00, -2.08366072e+01, -5.79206442e+00, -1.51489045e+01,
       -7.98407864e+00, -2.22670032e+00, -1.29231647e+01, -1.42250353e+01,
       -4.85098424e+00, -1.55022167e+01, -1.16895780e+01, -1.13297113e+01,
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        4.47745006e+00,  1.20014404e+01,  9.26638718e+00,  1.72278613e+01,
        1.28242058e+00,  4.97313483e+00,  2.70886861e+00,  1.06325767e+01,
        1.99776509e+01,  1.33595285e+01,  6.69811157e+00,  1.97969626e+01,
        9.20675671e+00,  5.38511196e+00,  1.09207933e+01,  9.15056287e+00,
        1.38958130e+01,  8.37821198e+00,  1.29146071e+01,  7.75851683e+00,
        2.07560668e+01,  1.25093297e+01,  1.21127769e+01,  1.02526090e+01,
        6.54569097e+00,  1.54901590e+01,  1.91511415e+01,  5.23096840e+00,
        1.27491930e+01,  1.94121891e+01,  2.10143858e+01,  1.04244176e+01,
       -4.01407089e+00,  6.12832562e+00,  1.61433969e+01,  7.44783864e+00,
        1.49092057e+01,  1.27183198e+01,  7.42188390e+00,  6.98395866e+00,
        1.24115704e+01,  1.61740263e+01,  1.00998594e+01,  1.38181689e+01,
        8.15682975e+00,  1.18222445e+01,  1.64897864e+01,  1.24620439e+01,
        8.94944574e+00,  8.02529813e+00,  4.84399737e+00,  7.48967620e+00,
        7.51276902e+00,  1.52264139e+01,  7.45670520e+00,  1.11922573e+01,
        1.69352517e+01,  1.18772722e+01,  1.38676541e+01,  8.82911505e+00,
        5.76996395e+00,  8.12437321e+00,  1.15654340e+01,  1.08886593e+01,
        2.06759318e+01,  5.05932287e+00,  1.03608856e+01,  9.93496930e+00,
        8.90035469e+00,  1.23938253e+01,  1.97303162e+01,  5.07660111e+00,
        5.61576909e+00,  2.97384139e+00,  1.42071844e+01,  7.81655808e+00,
        2.44912367e+01,  1.55247452e+01,  1.83196069e+01,  1.31136718e+01,
        1.77465912e+01,  6.13688072e+00,  5.40102339e+00,  9.07984482e+00,
        8.03918827e+00,  2.02061965e+01,  6.82066108e+00,  9.94425953e+00,
        6.16797652e+00,  5.73287004e+00,  1.68680797e+01,  8.67054696e+00,
       -4.61205243e+00,  4.19296727e+00,  1.64764601e+01,  3.07452431e+00,
        1.46407121e+01,  1.36268645e+01,  6.80205646e+00,  1.47669654e+01,
        1.34824296e+01,  1.27625826e+01,  1.00288293e+01,  1.22182911e+01,
        1.25763959e+01,  1.04193590e+01,  5.61013474e+00,  1.77682873e+01,
        1.10944471e+01,  3.59161073e+00,  1.00182434e+01,  9.36054391e+00,
        2.69092226e+00,  1.28946463e+01,  9.66476562e+00,  1.12441923e+01,
        6.68957691e+00,  1.08146574e+01,  1.22006859e+01,  1.45289794e+01,
        1.87496279e+01,  1.70786164e+01,  1.05703775e+01,  9.61575237e+00,
        1.33710957e+01,  9.01759573e+00,  2.10693580e+00,  7.33650306e+00,
        1.06594693e+01,  1.67931087e+01,  1.40235878e+01,  9.56131704e+00,
        1.39451773e+01,  7.13844853e+00,  7.25073885e+00, -2.94529585e+00,
        1.85572891e+01,  1.08482009e+01,  5.79805409e+00,  1.15949590e+01,
        1.03792652e+01,  1.04497678e+01,  6.69246486e+00,  1.37753970e+01,
        1.28698423e+01,  1.43046111e+01,  8.22800618e+00,  1.10627519e+01,
        8.27093437e+00,  1.62615199e+01,  1.03964329e+01,  1.50819496e+01,
        1.06418231e+01,  6.53750297e+00,  1.24581311e+01,  1.09995041e+01,
       -3.89564670e-01,  7.38437572e+00,  1.19238790e+01,  1.09655686e+01,
        8.19007220e+00,  1.20480641e+01,  8.14878459e+00,  9.67935940e+00,
        6.34900262e+00,  2.11399868e+01,  1.75967183e+01,  7.79060770e+00,
        6.96877863e+00,  1.31671970e+01,  1.63243087e+01,  1.96657006e+01,
        1.77101236e+01,  8.35264918e+00,  9.59649322e+00,  8.53155505e+00,
        6.09217025e+00,  4.50676667e+00,  3.50590246e+00,  1.00049962e+01,
        8.51712958e+00,  2.05044008e-01,  1.52216363e+01,  1.89633204e+01,
        8.86436091e+00,  1.03311471e+01,  5.98170284e+00,  1.32419705e+01)

hist(dataset_one, col=rgb(0,0,1,1/4),xlim=c(-30,30), breaks=80)
hist(dataset_two,  col=rgb(1,0,0,1/4), add=T, breaks=80)

legend("topright", c("Dataset 1", "Dataset 2"), fill=c("blue", "red"))
box()

variance <- function(x) mean((x-mean(x))^2)

print(paste("The mean of dataset_one is", mean(dataset_one)))

[1] "The mean of dataset_one is 0.3133598953587"

print(paste("The mean of dataset_two is", mean(dataset_two)))

[1] "The mean of dataset_two is 0.05276583677259"

print(paste("The variance of dataset_one is", variance(dataset_one)))

[1] "The variance of dataset_one is 101.080492211241"

print(paste("The variance of dataset_two is", variance(dataset_two)))

[1] "The variance of dataset_two is 124.745183427804"