In the context of sampling, Bessel’s Correction improves the estimate of standard deviation: specifically, while the sample standard deviation is a biased estimate of the population standard deviation, the bias is smaller with Bessel’s Correction.

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For data, we will use the diamonds data set in the R-Package ggplot2, which contains data from 53940 round cut diamonds. Here are the first 6 rows of this data set:

## # A tibble: 6 x 10
##   carat       cut color clarity depth table price     x     y     z
##   <dbl>     <ord> <ord>   <ord> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
## 1  0.23     Ideal     E     SI2  61.5    55   326  3.95  3.98  2.43
## 2  0.21   Premium     E     SI1  59.8    61   326  3.89  3.84  2.31
## 3  0.23      Good     E     VS1  56.9    65   327  4.05  4.07  2.31
## 4  0.29   Premium     I     VS2  62.4    58   334  4.20  4.23  2.63
## 5  0.31      Good     J     SI2  63.3    58   335  4.34  4.35  2.75
## 6  0.24 Very Good     J    VVS2  62.8    57   336  3.94  3.96  2.48

Describing the distribution of the “price” variable

Answer this question: what is the meaning of a distribution of a variable, and how does it relate to price?

The distribution of a variable describes what a variable takes and how often it takes those values.

Type of variable chosen

Explain what a quantitative variable is, and why it was important to make such a choice in a report about standard deviation. Explain how the concepts of numerical and quantitative variables are different, though related.

A quantitative variable is a variable measured by numbers. Numbers are critical to deducing the standard deviation, and that is their relation. Numerical data is data using numbers that it does not make sense or is not useful to add or average. An example would be social security numbers.

Histogram of diamonds price.

What is a histogram? Explain graph below.

The histogram is used to show the distribution of variables with rectangles in bins. Each bin in the graph below spans $750 and shows the amount of diamonds that are within that price range.

Violin plot

Explain the relationship between a histogram and a violin plot.

Violin plots are another way of representing what is shown in a histogram; it is a histogram on its side, reflected.

Numerical Summaries

R has a function that returns numerical summaries of data. For example:

summary(diamonds$price)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     326     950    2401    3933    5324   18823

Describe what each of these numbers means.

The lowest number in the set; median of lower half of data; median; median of upper set; highest value.

Modified Box Plots

Describe the relationship of the numbers above to the modified box plot, here drawn inside the violin plot. Explain the difference between a boxplot and a modified box plot. Explain what an outlier is, and how suspected outliers are identified in a modified box plot.

The dots on the modified boxplot are outliers. The modified boxplot limits the maximum and minimum, and then represents those numbers with dots, where the regular boxplot does not.

Adding the mean to the plot

Add one sentence to indicate where the mean is on this plot.

The mean is the red dot.

Standard Deviation: Formulas

Explain the formulas below, say which uses Bessel’s correction.

The top formula includes Bessel’s correction, which is supposed to make the standard deviation more accurate.

\[s = \sqrt{\frac{1}{n-1}\sum\left(x_i - \bar x\right)^2}\]

\[s_n = \sqrt{\frac{1}{n}\sum\left(x_i - \bar x\right)^2}\]

Standard Deviation of Diamonds Price

We compute the standard deviation (with Bessel’s correction) of the price variable:

sd(diamonds$price)
## [1] 3989.44

How about without Bessel’s correction? Well, R doesn’t seem to have this function, but we can add it:

sdn <- function(x) {
  return(sqrt(mean((x - mean(x))^2)))
}
sdn(diamonds$price)
## [1] 3989.403

How close are these estimates? Which is larger?

The Bessel’s correction number is larger, but it makes little difference because the values are nearly identical.

So what is the big deal about Bessel’s correction? See below.

Sampling

The statement that began this document asserted that Bessel’s correction is important in the context of sampling. Explain sampling here: explain the differences between a population, and a sample, and between a parameter and a statistic. Give examples of parameters and give examples of statistics. Explain the difference between the sample mean and the population mean. Explain the difference between the sample standard deviation and the population standard deviation.

A population is the total number of subjects in a particular group. A sample is used to estimate the values of a population, and it is a fraction of a population. Statistics describe samples while parameters describe the population. The sample standard deviation is the standard deviation within a sample.

We can sample from the diamonds data set and display the price of the diamonds in the sample.

Sample size, \(n\).

First, we need to choose a sample size, \(n\). We choose \(n=4\) which is very low in practice, but will serve to make a point.

sample.size <- 4

Set the seed of the pseudorandom number generator.

Sampling is random, so next we set the seed. Explain what a seed of a random number generator is. Explain what happens when you use the same seed and what happens when you use different seeds. The simulations below may help you.

The seed tells the computer which numbers to display, and is not random although it appears to be. Different seeds show different numbers

set.seed(1)

Sample once and repeat.

Now let’s try sampling, once.

sample(diamonds$price, sample.size)
## [1] 5801 8549  744  538

Explain what this command did.

The command selected 4 numbers from the set produced by the seed.

Let’s try it with another seed:

set.seed(2)
sample(diamonds$price, sample.size)
## [1] 4702 1006  745 4516

And another:

set.seed(3)
sample(diamonds$price, sample.size)
## [1] 4516 1429 9002 7127

And back to the first one:

set.seed(1)
sample(diamonds$price, sample.size)
## [1] 5801 8549  744  538

Explain these results.

When the seed changes, the numbers change.

Finally, what happens when we don’t set a seed, between samples.

set.seed(1)
sample(diamonds$price, sample.size)
## [1] 5801 8549  744  538
sample(diamonds$price, sample.size)
## [1] 4879 1976 2322  907
sample(diamonds$price, sample.size)
## [1]  463 3376 4932 4616
set.seed(1)
sample(diamonds$price, sample.size)
## [1] 5801 8549  744  538
sample(diamonds$price, sample.size)
## [1] 4879 1976 2322  907
sample(diamonds$price, sample.size)
## [1]  463 3376 4932 4616

Explain these results.

Without a seed, the samples will be produced with random numbers within the seeded set.

Describing samples with one number: a statistic

set.seed(1)
mean(sample(diamonds$price,sample.size))
## [1] 3908
mean(sample(diamonds$price,sample.size))
## [1] 2521
mean(sample(diamonds$price,sample.size))
## [1] 3346.75

Explain what we have done here. Answer the following question: what other statistics could we use to describe samples?

We have the mean of the seeded samples here. Other measurements would be standard deviation, median, and the quartile ranges.

For example standard deviation, with Bessel’s correction:

set.seed(1)
sd(sample(diamonds$price,sample.size))
## [1] 3936.586
sd(sample(diamonds$price,sample.size))
## [1] 1683.428
sd(sample(diamonds$price,sample.size))
## [1] 2036.409

And standard deviation, without Bessel’s correction:

set.seed(1)
sdn(sample(diamonds$price,sample.size))
## [1] 3409.183
sdn(sample(diamonds$price,sample.size))
## [1] 1457.891
sdn(sample(diamonds$price,sample.size))
## [1] 1763.582

Sampling Distributions of Statistics

Explain what a sampling distribution of a statistic is and how it relates to the numbers computed above. Answer the following question: what tools do we have to describe these distributions?

It is a representation of the values of a statistic using max, min, outliers, median, mean, etc. The numbers above can be used to describe our sample.

Sampling distribution for the mean of price of a sample of diamonds.

The plot below shows images of the sampling distribution for the sample mean, for different values of sample size.

Answer the following questions: what do the features of the graph below represent? One hint: the horizontal line is the population mean of the prices of all diamonds in the data set.

The box plots show the max, min, outliers, quartiles, medians

Explain the concept of an estimator. What is the sample mean estimating, and it what situation does it do a better job?

An estimator gives us an idea of some part of the sample. The sample mean estimates what the mean of the population will be.

Let’s try describing the sampling distribution of the sample standard deviation with Bessel’s Correction. Again the samples are of diamonds, and the variable considered is the price of diamonds:

Some people argue that it is appropriate to drop Bessel’s correction for populations, but if the population size is large, as shown here it doesn’t matter much. Why? What is the sample standard deviation estimating? In what situations is it a better estimate?

Now let’s try without Bessel’s correction:

Answer the following questions: what is the difference between the standard deviation with Bessel’s correction and the standard deviation without Bessel’s correction? Which do you think is better and when does it matter?

Bessel’s correction seems to represent the population mean better

Sampling error and sampling bias

Describe the difference between sampling error and sampling bias. Describe the difference between a biased estimator and unbiased estimators.

Sampling error is the difference between the statistic and the parameter i.e. sample and population, sampling bias is the mean sampling error