Histograms and Distributions

So far, we have focused on single estimates. How much volume is this tank, how long is this room, etc. Now we ask questions about several objects at once.

  • How tall are the students in this class?
  • How big are the classes at my university?
  • How big are the departments at my university?

Here we are interested in the behavior of a single variable. Is it evenly distributed or are some values more present than others? What are the largest and smallest values?

Characterizing a body of data

We often ask two questions about data. The first is what is the central tendency or do the data point to some sort of majority value. The second is how tightly clustered are the data. Does the data range widely?

Mean or Average

A common measure of the central tendency is the mean. If we take a list of data, sum the values, and divide by the number of data we get the mean or the average.

Median

Another measure of the central tendency is the median. The median is the value at which half the data has a value below the median and half the data has a value above it.

Standard Deviation

The standard deviation is a measure of how dense or spare the data are around the central tendency.

Histogram

A common tool for looking at single-variable (univariate) data is a histogram.

A histogram

  • Shows data of a single quantitative continuous variable
  • Shows the value of that data on the x-axis
  • Divides the x-axis into evenly spaced bins
  • On the y-axis shows the number of values in each bin
  • We should know what values are being counted
State Areas

Here we show how to go from raw data to a histogram.

State Populations Alphabetical

With the data sorted we can see the median value as the one in the middle.

State Populations Sorted
State Populations Histogram

Types of Data

  • Nominal data has no quantitative value.
    • Examples include state of birth, blood type, political affiliation
  • Ordinal data has a number and order but isn’t continuous
    • Example: survey question 1 for very unsatisfied, 5 for very satisfied
  • Continuous data: data that can take any value
    • Example: length, mass

Imposters

There are things that look like histograms but are not.

  • Time Series
  • Averaged Time Series
  • Bar charts with ordinal data

Percentile

The percentile takes the relative percentage and returns the absolute value corresponding to that relative percentage.

The median is the same thing as the 50th percentile. That is, the value that is greater than 50% of the data.

Percent Rank

The percent rank takes an absolute value and returns the relative position (the percentage of data below that value) within the distribution.

You could input your income and figure out if you are in the 99% or the 1% using percent rank.

Percentile and Percent Rank

Probabilities

These tools let you ask questions about how likely it is to observe events.

In the figure above, if you draw a random single sample from the population that the distribution represents, there is a 40% chance that it will be greater than the value represented by the vertical line.