Overview: Descriptive Statistics

1.     What are descriptive statistics?




2.     Descriptive vs inferential statistics




3.     Why the descriptive matter



4.     The “Big 7” descriptive statistics

Measures of central tendency

To make this a little more tangible, let’s look at a sample dataset, along with the corresponding
mean, median and mode. This dataset reflects the service ratings (on a scale of 1 – 10) from 15
customers.

Example set of descriptive stats

As you can see, the mean of 5.8 is the average rating across all 15 customers. Meanwhile, 6 is the
median. In other words, if you were to list all the responses in order from low to high, Customer 8
would be in the middle (with their service rating being 6). Lastly, the number 5 is the most
frequent rating (appearing 3 times), making it the mode.

To take this a step further, let’s look at the frequency distribution of the responses. In other
words, let’s count how many times each rating was received, and then plot these counts onto a bar
chart.

Example frequency distribution of descriptive stats

As you can see, the responses tend to cluster toward the centre of the chart, creating something of
a bell-shaped curve. In statistical terms, this is called a normal distribution.

As you delve into quantitative data analysis, you’ll find that normal distributions are very
common, but they’re certainly not the only type of distribution. In some cases, the data can lean
toward the left or the right of the chart (i.e., toward the low end or high end). This lean is
reflected by a measure called skewness, and it’s important to pay attention to this when you’re
analysing your data, as this will have an impact on what types of inferential statistics you can
use on your dataset.

Example of skewness









Measures of dispersion

Again, let’s look at our sample dataset to make this all a little more tangible.

Example of bell curve<br />

As you can see, the range of 8 reflects the difference between the highest rating (10) and the
lowest rating (2). The standard deviation of 2.18 tells us that on average, results within the
dataset are 2.18 away from the mean (of 5.8), reflecting a relatively dispersed set of data.

For the sake of comparison, let’s look at another much more tightly grouped (less dispersed)
dataset.

Example of skewed data

As you can see, all the ratings lay between 5 and 8 in this dataset, resulting in a much smaller
range, variance and standard deviation. You might also notice that the data are clustered toward
the right side of the graph – in other words, the data are skewed. If we calculate the skewness for
this dataset, we get a result of -0.12, confirming this right lean.

In summary, range, variance and standard deviation all provide an indication of how dispersed the
data are. These measures are important because they help you interpret the measures of central
tendency within context. In other words, if your measures of dispersion are all fairly high
numbers, you need to interpret your measures of central tendency with some caution, as the results
are not particularly centred. Conversely, if the data are all tightly grouped around the mean
(i.e., low dispersion), the mean becomes a much more “meaningful” statistic).

5.     Key takeaways

We’ve covered quite a bit of ground in this post. Here are the key takeaways:
  * Descriptive statistics, although relatively simple, are a critically important part of any
quantitative data analysis.
  * Measures of central tendency include the mean (average), median and mode.
  * Skewness indicates whether a dataset leans to one side or another
  * Measures of dispersion include the range, variance and standard deviation