DS2 – Displaying and Interpreting Single Data Sets

Basic Concepts: Construct displays including sector graphs, dot plots, radar charts, frequency tables, histograms and polygons Divide data sets into deciles, quartiles and percentiles Construct and use five number summaries and box and whisker plots Interpret data sets and identify misleading data

Dot Plots A dot plot is a column or row of dots. The number of dots is the frequency of the variable.

Sector graphs A sector graph presents data as sectors of a circle. The steps to construct a sector graph are: 1 Draw a circle and mark the centre. 2 Multiply the proportion of the whole by 360 to determine the sector angle. 3 Use a protractor to draw the angle with the vertex at the centre of the circle. 4 Repeat steps 1 and 2 until all the sectors or parts have been drawn. 5 Label all sectors.

Connor earns $600 and spends $240 on rent, $180 on food, $120 on petrol and saves $60. Construct a sector graph to represent this data. Connor ‘s expenditure

Divided Bar Graph A divided bar graph presents data as bars or rectangles. Steps to construct a divided bar graph are: 1 Draw a rectangle using an appropriate scale. 2 Multiply the proportion of the whole by the total bar length. 3 Use a ruler and draw the bar. 4 Repeat steps 1 and 2 until all the bars or parts have been drawn. 5 Label all bars or create a legend.

Radar Charts A radar chart (or spider chart) is used to compare the performance of one or more entities. Data on temperature, rainfall, humidity and sales are commonly presented using radar charts. 1 Determine the data to be presented as sectors. Draw the sectors. 2 Choose an appropriate scale for the data. Draw the scale beginning at the centre. 3 Draw line segments for each scale to create the ‘spider web’. 4 Plot the points and join them with a straight line. Create a legend if necessary.

Frequency Tables A frequency table (or frequency distribution) presents data so it can be more easily read. 1 Scores or outcomes are listed in the first column in ascending order. 2 Tally column records the number of times the score occurred (groups of 5s to make it easier to count). 3 Frequency column is the total count of each outcome.

Cumulative Frequency The cumulative frequency of each score is found by adding the frequencies of all the scores up to and including that particular score. The relative frequency of each score is the fraction of times that the score occurs. i Add a cumulative frequency column. ii Add a relative frequency column. b How many packets contained: i 40 or fewer Candy Clicks? ii fewer than 38 Candy Clicks? c What fraction of packets contained: i 37 Candy Clicks? ii 40 Candy Clicks? d What percentage of packets contained: i 38 Candy Clicks? ii 39 Candy Clicks?

Grouped data Consider the following data set…it is much easier to group the data than use every possible score. These two tables present the data above. Things to make sure of: the classes do not overlap. there are no gaps between class intervals. each class interval is the same size.

Frequency Histograms and Polygons A histogram is a column (or bar) graph in which the values of the variable are placed on the horizontal axis and the frequency of the variable on the vertical axis. There are no gaps between the columns. A frequency polygon is a line graph with the first and last points joined to the horizontal axis to form a polygon. The maximum temperature on each day in September was recorded, and the results summarised in a frequency table, as shown. a Draw a frequency histogram. b Draw a frequency polygon of the distribution.

Cumulative Frequency Histograms and Ogives A cumulative frequency histogram is a histogram with cumulative frequency on the vertical axis. A cumulative frequency polygon is a line graph formed by joining the upper right-hand corners of each column of the cumulative frequency histogram. A cumulative frequency polygon is also called an ogive. The maximum temperature, in °C, on each day in September, given in an earlier example is shown in the table. Draw a cumulative frequency histogram and a cumulative frequency polygon. Start by adding a cumulative frequency column. When drawing a cumulative frequency polygon, we join the upper-right hand vertices, not the midpoints of each column.

Range and Interquartile Range To analyse graphs we need to be able to find more information. A measure of the spread is calculated to determine whether most of the values are clustered together or stretched out. The range and interquartile range are measures of spread. The range is the difference between the highest and lowest scores. It is a simple way of measuring the spread of the data. The interquartile range does not rely on the extreme values like the range. The data is arranged in increasing order and divided into 4 equal parts or quartiles. The interquartile range (IQR) is the difference between the first (lower)quartile and third (upper)quartile. The first quartile cuts off the lowest 25% and the third quartile cuts off the lowest 75%.

Deciles, Quartiles and Percentiles

Using Cumulative Frequency Histograms and Ogives a Construct a cumulative frequency histogram and polygon or ogive. b Estimate the median using the ogive. There are 25 students, so the median student is 12.5. Draw a horizontal line from 12.5 until it intersects the ogive. Draw a vertical line from this point to the horizontal axis. Estimate the median value - The median is about 60.

Five number summaries A five number summary consists of five important statistics – lower extreme (or lowest value), lower quartile (or first quartile), median, upper quartile (or third quartile) and the higher extreme (or highest value). A five number summary can be read from various graphs, or used to construct a ‘Box and Whiskers’ plot. The final results of a survey are shown below. Construct a box-and-whisker plot, given the five-number summary: Minimum = 50, Q1 = 64, Median – 75.5, Q3 = 83 and Maximum = 95

Use a calculator for five number summaries! Press ‘mode’ ‘2’ for stats mode. Now the following screen will be displayed. Now press ‘1’ Check that you have a frequency column displayed. If you don’t have this, press then the button for setup and select ‘3’ then ‘1’ for ‘frequency on’. Enter each score by pressing the value followed by =. The calculator automatically makes the frequencies 1. Use the replay button to move to the frequency column. Highlight each frequency, one at a time, then type the appropriate number followed by =. When you are finished, press ‘AC’. Press ‘shift’ ‘1’’. This brings up a menu where you can select ‘5’ (Minmax): this provides information about the minimum and maximum scores, the median as well as the upper and lower quartiles. Select the one you want then press ‘=‘.

Stem-and-leaf plots Stem-and-leaf plots are another way of displaying information. They are used to group and rank data to show the range and distribution. The leaf is the final digit of a number, the preceding digits form the stem. The results in a mathematics class test are given below. a Draw a stem-and-leaf plot to represent this data. b What are the lowest, the median and the highest scores? c How many students scored: i 46 marks? ii 50 marks? iii 70 marks? iv a mark in the sixties? The leaves are put into ascending numerical order.

For each question, ALSO Make a five number summary from the stem and leaf plot!

Misleading graphs Misleading graphs give a false impression of the data, either by mistake or deliberately. The main causes of graphs being misleading are: • The scale on the vertical axis does not start at zero. • The scale on the vertical or horizontal axis is irregular. • The scale on the vertical or horizontal axis is missing. • The use of area or volume creates a false impression.

The table shows the profits of a company over the 5-year period from 2008 to 2012. Graphs A to E below are five ways of presenting this information graphically. Which features, if any, are misleading? Graphs A, B and D are fair, although each gives a different impression: Graph B makes the increase in profit look smaller as the horizontal scale is enlarged. Graph D makes the increase in profit look larger as the horizontal scale is compressed. Graphs C and E are misleading: Graph C has exaggerated the increase in profit by not starting the vertical scale at 0, thus enlarging this scale. Graph E has an irregular vertical scale, so the graph is incorrect.

Title text