Representing data - created from Mind Map

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Bar charts and pie charts

Bar charts

Bar charts are a simple but effective way of displaying data.

A bar chart consists of a series of bars of the same width, drawn either vertically or horizontally from an axis.

The heights (or lengths) of the bars always represent the frequencies.

The bars are generally separated by narrow gaps of equal width.

e.g. The frequency table shows the number of text messages received by a a group of students on a particular Sunday. The bar chart represents this information.

Line plots

A line plot is used to display small sets of discrete or categorical data. It is similar to a bar chart with dots (⬤) or crosses (✖) used instead of a bar. Each dot represents one unit of the variable.

The line plot below shows the sales of the different sized shoes by a shoe shop on a particular day.

The number of pairs sold was:

3 + 2 + 3 + 6 + 5 + 3 + 6 + 3 +1 = 32.

Pie charts

A pie chart is a good way of displaying data when you want to show how something is shared or divided. It is particularly suitable for displaying categorical data.

The pie chart is divided up into sectors.

e.g. The whole circle represents 24 hours.

We can see that Shane spent roughly the same time sleeping as he did at school.

The angle of each sector is proportional to the frequency of the category that it represents.

Histograms

One of the most common ways of representing a frequency distribution is by means of a histogram.

Histograms are very similar to bar charts but there are some important differences:

there are no gaps between the bars in a histogram.

histograms are used to show continuous data

the data is always grouped; the groups are called classes

the area of each bar or rectangle represents the frequency.

Histograms may have equal or unequal class intervals.

For our course we will confine our study to histograms with equal class intervals.

When the class intervals are equal, drawing a histogram is very similar to drawing a bar chart.

e.g. The frequency table below shows the times taken by 32 students to solve a problem.

The shape of a distribution

Histograms can be balanced (symmetrical) or skewed in some way.

Histograms are very useful when you want to see where the data lies and to get a clear picture of the shape of the distribution.

There are some shapes that occur frequently in distributions and you should be able to recognize and name them.

Symmetrical distribution

This distribution has an axis of symmetry down the middle. It is called a symmetrical distribution.

It is one of the most common and most important distributions in statistics. It is generally referred to as the normal distribution.

Real-life examples of a symmetrical (or normal) distribution are:

The heights of a random sample of people.

The intelligence quotient (IQ) of a population.

Positive skew

When a distribution has most of the data at the lower values, we say it has a positive skew. The following histogram shows a positive skew as most of the data represented by the higher bars, are mainly to the left.

Notice that there is a long tail to the right of the distribution.

N.B. If there is a positive skew, most of the data is to the left.

Negative skew

When a distribution has most of the data at the higher values, we say that the distribution has a negative skew. When a distribution has a negative skew the tail will be to the left.

In a distribution with a positive skew, the tail is to the right; with a negative skew, the tail is to the left.

Real life examples of a distribution with a negative skew are:

1. The ages at which people have to get their first pair of reading glasses.

2. The heights of players playing in a basketball league.

Stem and leaf diagrams

A stem and leaf diagram is a very useful way of representing data. It is useful because it shows all the original data and also gives you the overall picture or shape of the distribution.

It is similar to a horizontal bar chart, with the numbers themselves forming the bars.

Stem and leaf diagrams are suitable only for small amounts of data.

Often the stem shows the tens digit of the values and the leaves show the units digit.

If you put them together you get the original value.

You must always add a key to show how the stem and leaf combine.

N.B. If there is an uneven number of values, the middle value is 1/2(no. of values + 1).

e.g. If there are 30 values, the middle value is 1/2(30+1)=15.5. This will be half the sum of the 15th and 16th values.

Different values for the stems

Values that contain decimals can be represented on a stem and leaf diagram.

The units are the stem and the tenths are the leaves.

Back-to-back stem and leaf diagrams

Two stem and leaf diagrams ca be drawn using the same stem. These are known as back-to-back stem and leaf diagrams.

The leaves of one set of data are put to the right of the stem. The leaves of the other set of data are put on the left.

A back-to-back stem and leaf diagram is very useful to compare two sets of data.

Finding the interquartile range from a stem and leaf diagram

We found that the lower quartile is the value in the data that is one quarter way through the distribution. The upper quartile is the value that is three quarters way through the distribution. The difference between the upper quartile and the lower quartile is the interquartile range.

Scatter graphs

This section shows you how to compare two sets of data to find out if a relationship exists between them.

e.g. You could expect that a relationship exists between the number of ice-creams sold in a seaside shop and the average daytime temperatures.

Jane and her friend collected the following data to find out.

They plotted each pair of values, (10,1), (12, 5), (13, 15) and so on, on graph paper. They used the horizontal axis for the temperature and the vertical axis for the numbers of ice-creams sold.

Data such as (10, 1) and (12, 5) which comes in pairs is called bivariate data.

The points plotted above are called scatter diagram or scatter graph.

The diagram shows the that the number of ice-creams sold increases as the temperature increases. This indicates that there is an association or relationship between the temperature and the number of ice-creams sold.

If the points on a scatter graph lie approximately on a straight line, we say that there is a linear relationship between the two sets of data. The closer the points are to a straight line, the stronger the relationship will be.

Correlation

The strength of the relationship between two sets of data is known as correlation. If the points on a scatter graph lie on or close to a straight line, a strong correlation is said to exist.

Measuring correlation

The points on the scatter graph below are in a straight line. In this case we say that there is perfect positive correlation between the two variables.

If we use the letter r to represent the correlation and if perfect positive correlation exists, we say that r = 1.

In the diagram below, perfect negative correlation exists and r = -1.

All other correlations will have values between 1 and -1. If there is no correlation, r = 0.

The value of r is called the correlation coefficient.

The scatter diagrams below give examples of some values of r.

Causal relationships and correlation

When a change in one variable causes a change in another variable, we say that there is a causal relationship between them.

Correlation does not necessarily mean that there is pair of reading glasses.a causal relationship.