9.03 The shape of data

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.

The normal distribution looks like this bell-shaped curve.

This picture shows how a data set that has an approximate normal distribution may appear in a histogram.

The dark line shows the nice, symmetrical curve that can be drawn over the histogram that the data roughly follows.

In this distribution, the peak of the data represents the mean, the median and the mode (taken as the centre of the modal class) all these measures of central tendency are equal for this symmetrical distribution.

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 positively skewed has this general shape with right side stretched out.

General shape shown over a histogram of positively skewed data.

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.

A negatively skewed graph has this general shape with left side stretched out.

General shape shown over a histogram of negatively skewed data.

Examples

Example 1

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

a

What was the median age?

Worked Solution

Create a strategy

Find the middle score.

Apply the idea

We know that there 19 values recorded, so the 10th value will be the median.\text{Median}=20 \text{ years old}

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b

What was the difference between the lowest age and the median?

Worked Solution

Create a strategy

Subtract the lowest value recorded from the median.

Apply the idea

The lowest age recorded was 10 years old.

\displaystyle \text{Difference}	\displaystyle =	\displaystyle 20-10	Subtract the values
	\displaystyle =	\displaystyle 10 \text{ years}	Evaluate

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c

What is the difference between the highest age and the median?

Worked Solution

Create a strategy

Subtract the median from the highest value recorded.

Apply the idea

The highest age recorded was 54 years old.

\displaystyle \text{Difference}	\displaystyle =	\displaystyle 54-20	Subtract the values
	\displaystyle =	\displaystyle 34 \text{ years}	Evaluate

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d

What was the mean age? Round your answer to two decimal places if needed.

Worked Solution

Create a strategy

Use the formula: \text{Mean}=\dfrac{\text{Sum of scores}}{\text{Number of scores}}

Apply the idea

We can add the scores up in our calculator to get: 432.

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{432}{19}	Use the formula
	\displaystyle \approx	\displaystyle 22.74 \text{ years}	Evaluate the division

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e

Is the data positively or negatively skewed?

Worked Solution

Create a strategy

If most of the scores are relatively low, the distribution is positively skewed whereas if most of the scores are relatively high, the distribution is negatively skewed.

Apply the idea

Most of the scores appear in Stem 1 and Stem 2, so they are mostly low. So the distribution is positively-skewed.

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Example 2

State whether the scores in each histogram are positively skewed, negatively skewed or symmetrical (approximately).

a

Worked Solution

Create a strategy

Check where the bulk of the data sits and look at the general shape of the distribution.

Apply the idea

The scores are roughly even in both the high and low end, so the distribution is symmetrical.

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b

Worked Solution

Apply the idea

The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively-skewed.

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c

Worked Solution

Apply the idea

The scores have higher-frequency columns on the right (the higher scores), so the distribution is negatively-skewed.

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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.

If the data has two clear peaks then the shape is called bimodal.

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.

For the dot plot given above the score of 9 would be considered an outlier as it is well above the body of the data.

Examples

Example 3

The percentage of faulty computer chips in 42 batches were recorded in the histogram below.

a

Which of the following makes this statement true? The distribution is:

A

Uni-modal

B

Bi-modal

C

Multi-modal, but not bi-modal

Worked Solution

Create a strategy

Count the number of peeks in the distribution.

Apply the idea

Looking at the histogram, there are two peaks. So the distribution is bi-modal. The correct answer is option B.

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b

Which of the following are the modal classes? Select all that apply.

A

0-1

B

1-2

C

2-3

D

3-4

E

4-5

F

5-6

Worked Solution

Create a strategy

The modal classes are those with the highest frequency.

Apply the idea

The highest frequency is 12, and the classes with this frequency are 1-2 and 3-4. So the correct answers are options B and D.

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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.

Common shape of a symmetrical distribution

Histogram of approximately symmetrical data

Box plot of symmetrical 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.

General shape positively skewed data

Histogram of positively skewed data

Box plot of positively skewed data

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.

General shape negatively skewed data

Histogram of negatively skewed data

Box plot of negatively skewed data

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.

Examples

Example 4

Match the histograms to its box plot.

Worked Solution

Create a strategy

Look for characteristics of skewed and symmetric distributions of data.

Apply the idea

• Boxplot A and histogram 3 have long right tails, so they are both right skewed.

• Boxplot C and histogram 2 have long left tails, so they are both left skewed.

• Boxplot B and histogram 1 are both approximately symmetric.