Descriptive Statistics

Opening datasets

Sometimes pre-existing datasets are imported into R Studio. There are several different types of datasets that can be imported. The ones used most frequently in this course are from Excel and SPSS. Types of computer files are indicated by what follows the period (“.”) in the file name. For example file.docx is a type of Word file.

Here’s a list of common file types we’ll use.

• Excel = .xlsx
• SPSS = .sav
• Comma Separated Values = .csv

One file we’ll be using today is the Album Sales file. You can either import the file in R Studio by downloading it first or you can actually import the file from the website itself. So instead of importing the file on you computer you simply point R Studio to the web address of the file. To make things easier, that’s how we will do it here.

Here is the code for importing the file from the web.

library(haven)
Album_Sales <- read_sav("https://github.com/jvanster/Psy_300_Datasets/raw/refs/heads/main/SPSS_Datasets/Album%20Sales.sav")
View(Album_Sales)

Descriptive Statistics

Descriptive statistics are just what the name implies, they are used to describe a dataset. This is different from inferential statistics, which are used to infer population parameters based on a sample.

One of the most common sets of descriptive statistics is known as central tendency, which is simply finding a way to get a basic overview of a dataset as a whole.

Mean

One measure of central tendency is the mean or average, which shows us the center of a distribution of data. The formula for the mean is very straightforward. It’s simply the sum of all the scores divided by the number of scores. The symbol for the sample mean is an x with a bar over it. \[\bar x=\frac{\Sigma x}{n}\] The capital Greek letter Sigma in the numerator stands for sum and X stands in for all the values in a dataset with the N standing for the number of values in the dataset. In R Studio, we simply use the mean command to find the mean for a particular variable

mean(diamonds$carat)

[1] 0.7979397

Median and Mode

There are two other measures of central tendency that are often used in statistics, median and mode. The median is simply the center score, so to find it, you simply rank order your values from lowest to highest and the middle score in that rank order. For even scores you take the two middle scores add them and divide them by two. For odd scores there is only one middle score. Here again, R gives us an easy way to find the median.

median(diamonds$carat)

[1] 0.7

The mode is the score or value with the highest frequency in your dataset. So whatever value occurs the most times is the mode. For some reason, mode is not part of the basic R package, so you need to install the “modeest” package if you need to find this descriptive statistic.

library(modeest)

Once the library is installed use the mfv command

mfv(diamonds$carat)

[1] 0.3

We tend to use the mean most often for central tendency, but it’s more effected by extreme scores. So if you have several low or high outliers (extreme scores that are higher or lower than the mean), the median may be a more accurate representation of the data. More often than not, we’ll use the mean. The mode is a more straightforward measure. So if you were interested in the most popular song on an album based on downloads, you would find the mode. The mode is often very helpful with categorical data.

Variability

The second most important aspect of descriptive statistics is variability or dispersion in a dataset. This typically represents how the rest of the dataset relates to the mean or some other measure of central tendency. For example: are the scores close to or widely dispersed from the mean?

Standard Deviation

Often times we’re interested in the average spread or deviation from the mean. Deviance is the distance of any particular score from the mean. So to find the deviance you simply take the score and subtract it from the mean. \[deviance = \bar x - x\] If we wanted to find out the total amount of deviance, we could simply add together the total deviance for each number in our dataset. So the equation would look like this:

\[total\;deviance = \Sigma(X-\bar X)\]

Unfortunately this equation causes some problems. All the positive values above the mean and negative values below the mean cancel each other out, leaving a total deviance of zero. Zero doesn’t tell us much about variability and deviation, so we need a way to fix this issue.

The way to overcome this problem is by squaring each deviation score, which makes all the deviation values positive and thus produces a positive number. This number is called the sum of squared errors or the sum of squares (SS) with the following formula.

\[sum\;of\;squares (SS) = \Sigma(X-\bar X)^2\]

This number is still somewhat inflated, since it’s constructed based on squared values. One way to fix this issue to find the average dispersion, which will be based on the number in our sample. Since the sample is an estimate of the population, we actually don’t use N, but N -1. This number is called variance and has the following symbol and equation.

\[variance(s^2) = \frac {SS}{n-1} = \frac {\Sigma(x-\bar x)^2}{n-1}\]

This number is closer to the original units of measurement, but to make it more accurate, the original squared values need to be taken back out of the measure. To do this, the square root of the variance is calculated.

\[s = \sqrt {\frac {\Sigma (x- \bar x)^2}{n-1}}\]

To find the standard deviation using R Studio we use the sd command to find it.

sd(Album_Sales$Adverts)

[1] 485.6552

Range

Another measure of dispersion is range. This is simply subtracting the lowest score from the highest score for a particular variable or vector of scores. So to find the range simply use the range function and then subtract the two numbers provided, which are the min and max.

range(Album_Sales$Adverts)

[1]    9.104 2271.860

Then simply subtract the scores

2271.860-9.104

[1] 2262.756

Interquartile Range

Another helpful type of range is the interquartile range, which is the range of numbers from the 25th and 75th percentile. Percentiles are just dividing up a dataset based on where certain scores are based on percentages. For example, we can look at what score is at the 50th percentile. To do that we use the quantile function. x is the variable we are analyzing, and .5 is the percentile (.5 = 50%; .35 = 35%)

quantile(x = diamonds$carat, probs = .5)

50% 
0.7 

So the number at the 50th percentile is 0.7. Notice that this is the same as the median, described earlier. Remember that the median is the number from the dataset that is in the middle or center of the dataset. Because the mean is calculated, it may or may not be a number in the dataset.

So to find the interquartile range, we want our two quartiles or numbers at “quarters” of the dataset, so 25% and 75%. Think of a dollar, which is made up of 4 quarters. We already know that half a dollar is 50 cents, which would be the median, but the other two quartiles (quarters) would be 25 cents and 75 cents.

quantile(x = diamonds$carat, probs = c(.25, .75))

 25%  75% 
0.40 1.04 

Finally, to get the interquartile range we simply subtract these two numbers.

1.04-.40

[1] 0.64

So the interquartile range is 0.64.