When we describe the shape of data sets, we want to focus on how the scores are distributed. Some questions that we might be interested include:
Is the distribution symmetrical or not?
Are there any clusters or gaps in the data?
Are there any outliers?
Where is the centre of the data located approximately? (Recall our three measures of centre: mean, median and mode)
Is the data widely spread or very compact? (Recall our three measures of spread: range, interquartile range and standard deviation)
Data may be described as symmetrical or asymmetrical.
There are many cases where the data tends to be around a central value with no bias left or right. In such a case, roughly 50 \% of scores will be above the mean and 50 \% of scores will be below the mean. In other words, the mean and median roughly coincide.
The normal distribution is a common example of a symmetrical distribution of data.
A uniform distribution is a symmetrical distribution where each outcome is equally likely, so the frequency should be the same for each outcome. For example, when rolling dice the outcomes are equally likely, while we might get an irregular column graph if only a small number of rolls were performed if we continued to roll the dice the distribution would approach a uniform distribution like that shown below.
If a data set is asymmetrical instead (i.e. it isn't symmetrical), it may be described as skewed.
A data set that has positive skew (sometimes called a 'right skew') has a longer tail of values to the right of the data set. The mass of the distribution is concentrated on the left of the figure.
A data set that has negative skew (sometimes called a 'left skew') has a longer tail of values to the left of the data set. The mass of the distribution is concentrated on the right of the figure.
The stem-and-leaf plot below shows the age of people to enter through the gates of a concert in the first 5 seconds.
Stem | Leaf |
---|---|
1 | 0\ 1\ 2\ 3\ 4\ 5\ 6\ 6\ 6 |
2 | 0\ 0\ 1\ 4\ 9 |
3 | 1\ 4\ 7\ 9 |
4 | |
5 | 4 |
Key 1\vert 2 = 12 years old |
What was the median age?
What was the difference between the lowest age and the median?
What is the difference between the highest age and the median?
What was the mean age? Round your answer to two decimal places if needed.
Is the data positively or negatively skewed?
State whether the scores in each histogram are positively skewed, negatively skewed or symmetrical (approximately).
A distribution is said to be symmetric if its left and right sides are mirror images of one another.
A uniform distribution is a symmetrical distribution where each outcome is equally likely, so the frequency should be the same for each outcome.
A data set that has positive skew (sometimes called a 'right skew') has a longer tail of values to the right of the data set. The mass of the distribution is concentrated on the left of the figure.
A data set that has negative skew (sometimes called a 'left skew') has a longer tail of values to the left of the data set. The mass of the distribution is concentrated on the right of the figure.
In a set of data, a cluster occurs when a large number of the scores are grouped together within a small range. Clustering may occur at a single location or several locations. For example, annual wages for a factory may cluster around \$ 40\,000 for unskilled factory workers, \$ 55\,000 for tradespersons and \$ 70\,000 for management. The data may also have clear gaps where values are either very uncommon or not possible in the data set.
As we have seen previously, an outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations. Outliers are important to identify as they point to unusual bits of data that may require further investigation and impact some calculations such as mean, range, and standard deviation.
The percentage of faulty computer chips in 42 batches were recorded in the histogram below.
Which of the following makes this statement true? The distribution is:
Which of the following are the modal classes? Select all that apply.
To determine the modality of a data distribution:
If there is a single class the data is uni-modal.
If there are two classes the data is bi-modal.
If there are more than two the data is multi-modal.
An outlier is a value that is either noticeably greater or smaller than other observations.
These two displays are great for being able to identify key features of the shape of the data, as well as the range and in the case of the box plot, the interquartile range and median.
We should expect then that the shape of the data would be the same whether it is represented in a curve, box plot or histogram. Remember that the shape of data can be symmetric, negatively skewed or positively skewed.
For a symmetric distribution:
The median is in the centre of the range and the tails (whiskers) of the data are of equal length.
The graph should be approximately a mirror image of itself about the centre of the data.
For a positive skewed (skewed right) data distribution:
The data is stretched out to the right, producing a longer tail (whisker) to the right of the graph.
The bulk of the data is to the left. Higher frequency columns and the box should appear to the left.
For a negative skewed (skewed left) data distribution:
The data is stretched out to the left, producing a longer tail (whisker) to the left of the graph.
The bulk of the data is to the right. Higher frequency columns and the box should appear to the right.
Looking at the diagrams above, can you see the similarities in the representations?
We can see the skewed tails, where the bulk of the data sits and general shape. These are some of the features you can use to match histograms and box plots. We can also look at the data range.
Match the histograms to its box plot.
Symmetric
The median is in the centre of the range and the tails (whiskers) of the data are of equal length
The graph should be approximately a mirror image of itself about the centre of the data
Positive skewed (skewed right)
The data is stretched out to the right, producing a longer tail (whisker) to the right of the graph
The bulk of the data is to the left-higher frequency columns and the box should appear to the left
Negative skewed (skewed left)
The data is stretched out to the left, producing a longer tail (whisker) to the left of the graph
The bulk of the data is to the right-higher frequency columns and the box should appear to the right