Histograms

A histogram is an effective way to represent data. It consists of bars (or rectangles) where the base of the rectangle represents a range of numbers, and the height of the rectangle is the number of times the measured quantity occurs within that range.

Consider the following example which represents tests scores. The bars in the histogram represent the number of students who scored within the range at the base of the rectangle.

In the above example, we see that 17 students scored between 80 and 90 on the test. The total number of students who wrote the test could be determined by adding together the heights of all the bars.

Histograms can either be drawn so the bars are touching, or the bars can be separated by some white space.

Here are two histograms that would result from a seven question poll. The base of the bars is not labeled by a range any more, but by a number representing the seven questions. The number of people in the sample could be determined by adding the heights of all the bars, assuming that each member of the sample could only choose one of the seven possible responses.

Example

The Gov of Minnesota wants to have some idea what the residents of Minnesota think about the issue of raising taxes to improve education facilities (infrastructure) in the state.

Before serious discussions begin, they want to take the pulse of the residents by conducting a simple poll.

To do this, they decide to conduct a phone poll: they will phone, choosing phone numbers at random, 500 residents of Minnesota and ask them to choose the answer that most represents their position on the question "Should Minnesota raise state taxes to improve the infrastructure of the schools in Minnesota". The choices offered are:

1) Yes, if my taxes do not go up by more than $100.

2) No, most definitely not.

3) Undecided/need more information.

4) If you are going to raise taxes, spend the money on something else.

Minnesota has a population of almost 5 million people (census). This is the population.

I ran my own computer simulation, constructing a population of 5 million with a response to the question from 1 to 4, and then sampling the population at 500 randomly chosen positions.

The way I have constructed my data, I have ignored diversity which might play an important role in the real world application of this method. Some of the things that might make my study not sample diversity properly would be:

Rural vs urban residents--what area code are we calling most often?

What time of day the phone calls are made--if calls are made during the day, we would sample more heavily people who do not work during the day.

Do we hang up if we get an answering machine--what about people who screen their calls?

Since we are interested in seeing how accurate our sampling techniques are, here is the histogram for the entire population. This information is, of course, unknown to the person taking the poll.

20.8% 1) Yes, if my taxes do not go up by more than $100. 33.4% 2) No, most definitely not. 33.3% 3) Undecided/need more information. 12.5% 4) If you are going to raise taxes, spend the money on something else.

Here are the results from my ''poll'' of 500 people chosen randomly from the population:

18.8% 1) Yes, if my taxes do not go up by more than $100. 37.0% 2) No, most definitely not. 35.0% 3) Undecided/need more information. 9.2% 4) If you are going to raise taxes, spend the money on something else.

Notice that we actually get quite good results from using "only" 500 people. Well designed samples can produce good estimates for the behaviour of the population.

The margin of error can be estimated using the formula 100/Sqrt(500) ~ 4.47%. This means that our sample gives us results that comes within plus or minus 4.47% of the truth 95% of the time.

Obviously, a smaller sample size increases the margin of error. However, since there is a square root in the quick method for margin of error, to cut the margin of error in half we must increase the sample size by 4!