Descriptive Statistics

Describe the data we've collected using measures of center, measures of spread, shape of our distribution, and outliers. We can also use plots of our data to gain a better understanding.

Types of data

Reminder:

5 number summary

Q2 = median.

Q1 = median of the first half of the data.

Q3 = median of the 2nd half of the data.

Range = max - min

Interquartile range: Q3 - Q1.

These numbers and graph is a boxplot:

Spread of a dataset

Most common = standard deviation and variance.

Standard Deviation: On average, how much each point varies from the mean of the points. How to calculate?

Variance: Average squared distance of each observation from the mean

1. Find the mean. x-bar

2. Find the distance of each point of this mean: xi - x-bar

3. Square all the values (xi-x-bar)^2

This gives us the variance.

4. Take the square root of the variance = Standard deviation.

We initially squared to get positive values of xi-xbar. So to "cancel" it out, we take the square root.

Units

It,s important when comparing data that the units are the same. Ex: if measuring $, we'll use the std dev and not variance since std dev has same units are $.

The standard deviation is a measurement that has the same units as our original data, while the units of the variance are the square of the units in our original data. For example, if the units in our original data were dollars, then units of the standard deviation would also be dollars, while the units of the variance would be dollars squared.

Common usecase: find the std dev of different groups to find out which ones are more spread out.

Variance and Std Dev in Excel

To calculate the variance of a set of 10 values in a spreadsheet application, with our 10 data points in column A, we would create a new column B by typing in something like =A1-AVERAGE(A$1:A$10) and copying this down for all 10 rows. This would find us the difference between each data point and the mean average of all the data. Then we create a new column C having the square of these differences, using the formula =B1^2 in cell C1, and copying that down for all rows. Then in the cell below this new column, cell C11, type in =SUM(C1:C10). This adds up all these values in column C. Finally in cell C12, we divide this sum by the number of data points we have, in this case ten: =C11/10. This cell C12 now contains the variance for our 10 data points.

More detailed guidance on using spreadsheets like this may be included in a future lesson in your program.

In the same spreadsheet as above, to find the standard deviation of our same set of 10 data values, we would use another cell like C13 to take the square root of our variance measure, by typing in =sqrt(C12).

Shape

Mode = highest bar of the histogram.

Most common:

Real life examples are

Left skewed examples: Age of death, asset price changes, GPA

Right skewed examples: amount of drugs in your blood over time , distribution of wealth, human atheltic abilities.

Outliers

Understand the impact they have on our summary statistics.

If outliers are typos or mistakes = remove.

If not, why do they exist?

The median is the middle number and is not effected by outliers.

Usually the best way to see what's happenning is a visual.

Have to be careful of how we share our results.