Question

Draw any dotplot to show a dataset that is Clearly skewed to the left.

Short answer

The dotplot shows a clear left skew, with most of the data points clustered at the higher end (to the right) and fewer data points extended towards the lower end (to the left). The tail of this dotplot stretches to the left, indicating a left-skewed dataset.

Step-by-step solution

Identify Range of Data

First, identify the range of data which essentially includes the lowest and highest values. The range should be enough to illustrate the skewness of the data. For this exercise, assume the range as 1 to 10.

Draw Horizontal Axis Line and Scale

Draw a horizontal line on a piece of paper or software such as Excel or Google Sheets. This will serve as your axis line. Mark the range on this line - from 1 to 10. These are also called scale numbers.

Plot the Dots

Starting from the right (the larger numbers), begin to plot the dots above the scale numbers. When doing this, remember to place more dots above the numbers to the right, and fewer dots above the numbers on the left. The dots should gradually decrease in number as we move from right to left. This results in a left-skewed dotplot.

Review the Dotplot

Finally, review the dotplot. It should show a clear left skew, meaning there is a cluster of dots at the higher end of the scale and fewer dots towards the lower end.

Key concepts

Data Visualization

Data visualization is a critical skill in the world of statistics and analytics. It refers to the representation of data in a visual context, like a chart or graph, to help people understand the significance of data by placing it in a visual context. For students, visualizing data can transform complex datasets into a form that is easier to comprehend and interpret. Dotplots are a basic yet powerful form of data visualization particularly useful in displaying the distribution of a small set of quantitative data.

A well-designed dotplot can reveal the central tendency, variability, and skewness in data with just a simple glance. When creating a dotplot, each data value is represented by a dot above a number line, making it helpful for spotting patterns such as clusters, gaps, and outliers. This graphical tool is especially suitable for beginners because it requires minimal mathematical complexity while offering clarity into the nature of the dataset.

Statistical Skewness

Statistical skewness is a measure of the asymmetry in a distribution of data. Think of it as a way to describe which way and how much a dataset 'leans' or 'tails off'. There are three types of skewness: right (positive), left (negative), and none (zero).

In a left-skewed distribution, also known as negative skewness, the tail on the left side is longer or fatter than the right side. Therefore, the bulk of the values (including the median) are concentrated on the right. One useful tip for remembering this is that the skewness direction refers to the long tail of the distribution and not the peak. For students, recognizing the skewness from a dotplot involves looking at how the dots pile up towards one end and taper off towards the other end. In digitized datasets, calculating the skewness can be done with statistical software by looking at the sign and magnitude of the skewness coefficient.

Range of Data

The range of data is a simple measure of dispersion that identifies how spread out a set of numerical data is. It's calculated by subtracting the smallest value in the set from the largest one. The resulting number represents the length of the interval that contains all the data points.

Understanding the range is essential when creating a dotplot because it determines the scale of the number line, ensuring that all data points will fit on the graph. If the range is too small, it may not accurately reflect variations in data; if it's too large, the dotplot may become sparse and less informative. In our exercise, the range was set from 1 to 10 to accommodate all potential data points while effectively demonstrating the left skewness. Students should appreciate that careful consideration of the range enhances data visualization by aligning the detail and spread of the data with the scale of the graph.