Data Distribution Builder

What is a statistical question?

A statistical question is one that expects different answers, so you have to collect data. "How many books did each student in my class read this year?" is statistical, because every student will answer differently.

"How many books did I read this year?" is not statistical. There is only one answer. Statistics is about the spread of many answers, not a single fact.

What is this module?

The module shows one class data set: how many books each student read. The same set of numbers is drawn three ways at once: a dot plot, a histogram, and a box plot.

You can click to add a point, drag a point to change its value, and tap a point to remove it. The panel reports how many observations there are and the measures of center and spread. The sample buttons load data with different shapes.

What they're learning

The big idea is that a data set has a shape, and its center and spread describe that shape.

• Center. The mean and the median both try to give one typical value. The mean averages all the values; the median is the middle value.
• Spread. The range is the distance from smallest to largest. The interquartile range (IQR) is the spread of the middle half. The mean absolute deviation (MAD) is the average distance from the mean.
• Shape matters. Load the Outlier sample and drag the value 20. The mean races toward it while the median barely moves. When data is skewed by an outlier, the median is usually the more honest description.
• Three pictures of one data set. A dot plot shows every value, a histogram groups them into bins, and a box plot shows the five-number summary: minimum, Q1, median, Q3, maximum.

The Outlier sample is the whole lesson. If you only do one thing, load it and drag the far point.

Aliases

• Statistics, data analysis
• Measures of center and spread
• Mean, median, mode
• Dot plot, histogram, box plot
• Average

Refresher

A data set has two things worth knowing: a center, which is one typical value, and a spread, which is how far the values go. The mean and the median both describe the center. The range, the IQR, and the MAD describe the spread. When a data set has an outlier, the mean chases it and the median does not, so the median is often the more honest number.

Try this together

A few minutes at the screen is plenty.

1. Load the Symmetric sample. Point out that the mean and median are the same number, which happens when the shape is balanced.
2. Load the Outlier sample. Ask which number better describes a typical student, and why.
3. Drag the far point around and watch the mean follow it while the median stays put.
4. Count the dots in the box plot: it marks the smallest, the middle half, and the largest values.

Do not worry about memorising MAD or IQR. The goal is the idea that center and spread are two separate things.

Words you might hear

• Statistical question: a question that expects a range of answers and needs data to answer.
• Distribution: how the values are spread out, clustered, spread wide, piled on one side, or split into two peaks.
• Mean: the average. Add all the values and divide by how many there are.
• Median: the middle value when the data is put in order.
• Range: the largest value minus the smallest.
• Quartiles: Q1 and Q3 mark the quarter and three-quarter points. The box in a box plot spans them.
• Interquartile range (IQR): Q3 minus Q1, the spread of the middle half of the data.
• Mean absolute deviation (MAD): the average distance of the values from the mean. A small MAD means the data is tightly packed.
• Outlier: a value far from the rest. It pulls the mean but leaves the median alone.

Deeper dive

Start with a handful of numbers, say 2, 3, 3, 4, 20. Put them in order. The median is the middle one, which is 3. The mean is the total divided by how many there are, so 32 divided by 5, which is 6.4. The 20 is an outlier, and it is why the mean sits far above the median.

Spread is just how far the numbers reach. The range is the largest minus the smallest. The module draws the same set of numbers three ways: every value as a dot, values grouped into bars, and the five-number summary as a box.