2 Contents Revise measures of central tendency of ungrouped data Measures of central tendency in grouped dataRevision of Range as a measure of Dispersion and extension to include percentiles, quartiles, interquartile and semi-interquartile range.Five number summaries and box and whisker plotsUsing statistical summaries to make meaningful comments on the context associated with the given data.
3 Revision: Measures of Central Tendency of Ungrouped Data Mean: This is the average of a set of data. The mean is obtained by adding up all scores and then dividing the sum by the total number of scores.Mode: The mode is the most commonly occurring observation.\Median: The median is the middlemost score (for an odd number of scores) and the average of the middle two scores (for an even number of scores). The scores must be arranged in increasing order before the median can be determined.
4 Revision: Measures of Central Tendency of Ungrouped Data Eg: The following test scores were achieved by Sarah in her maths tests: 45% ; 60% ; 72% ; 65 % ; 60 % ; 88% ; 60% ; 73%.Mean:Mode: % (occurs 3 times)Median: (data must be ordered)45 ; 60 ; 60 ; 60 ; 65 ; 72 ; 73 ; 88
5 Measures of Dispersion Range: Highest score – lowest scoreQuartiles: points that divide the data set into quartersLower Quartile (Q1) – score at the first quarter sectionUpper Quartile (Q3) – score at the third quartile sectionInterquartile range: (Q3 – Q1)Semi-interquartile Range: ½ (Q3 – Q1)
6 Measures of Dispersion Eg: Given the following set of data:; ; ; ; ; ; ; ; 17Q3 = (16+16) ÷2Q1 = (7+8)÷2Median = Q2 =13SIQR = 16 – 7.5= 8.5Range = 17 – 7= 10
7 Approximate mean of grouped data: = 565÷26 = 21.73 IntervalFrequency (f)Midpoint (m)f x m0 ≤ x <10454x510≤ x <207157x1520≤ x <309259x2530≤ x <406356x35Total26565
8 Grouped Data cont.IntervalFrequency (f)Midpoint (m)f x m0 ≤ x <10454x510≤ x <207157x1520≤ x <309259x2530≤ x <406356x35Total26565Modal Class in Grouped Data = the class (group) with the highest frequency: 20≤ x <3To find the median, you need to draw an ogive.
9 5 Number Summary Minimum value Lower Quartile (Q1) Median (Q2) Upper Quartile (Q3)Maximum ValueA five number summary van be represented on a box and whisker plot.MinMaxQ1Q2Q3
11 Table of Contents: Use histograms to represent data. Use frequency polygons to represent data.Use pie diagrams to represent data.Use line and broken line diagrams to represent data
12 Data Handling involves: collecting data for a particular purposesorting out the datarepresenting the data in tables, charts or graphsanalysing the resultscoming to conclusions
13 Test Your KnowledgeWhat does data handling involve? A Collecting information, analyzing information and then representing it in a graph B Analyzing a frequency table and basing future decisions on this information C Making a decision and then calculating the averages to suit your decision
14 Use histograms to represent data. A histogram displays the frequency of either continuous (is used for continuous data like measurements ) or grouped discrete data in the form of bars.A histogram represents each class of information by means of a rectangle whose width represents the class width and whose height is proportional to the frequency.It is a “special” bar chart, but has no spaces between the bars / columns.
15 The reason the bars touch each other is because class intervals are drawn on a continuous line. Each bar represents the class frequency in a particular class and is mostly of equal width.Because we work with grouped data, we use the value of the midpoint of each class to represent the classRemember: in a histogram the columns are always touching each other and therefore we have to determine class boundaries and midpoints of intervals.
16 Test Your KnowledgeWhen would a learner use a histogram? A: For listed data B: For raw data C: For grouped data
17 Test Your KnowledgeThe table below shows the marks out of 100 in a maths test fora class of 32 students. Draw a histogram representing this data.
19 Use frequency polygons to represent data. A frequency table may be graphed in three ways:The histogramThe frequency polygon.The cumulative frequency curveA frequency polygon joins the midpoints of classes and although the histogram is shown it is not part of the frequency polygon.We need to know the midpoint of each class.The broken line segments joining these midpoints will give us a frequency polygon.
20 Test Your KnowledgeWhat does a frequency polygon join? A: Frequencies B: Midpoints of grouped data C: Joining dots of plotted points
21 Test Your KnowledgeFirst draw the histogram and then complete the frequency polygon.
23 Use pie diagrams to represent data. The graph is called a pie-graph, because of its shape. It is circular in shape and frequencies are represented by sectors. Each sector will give us a percentage of the total amount. Sectors can be given as percentages or degrees. A full circle represents either 100% or.
24 Use pie diagrams to represent data. Characteristics:a) The graph has a label, or heading, to indicate what itrepresents.b) All the frequencies are added and then written as afraction of the sum of frequencies.c) Each fraction can then be changed into a numberrepresenting percentages or degrees by multiplying by100% or 360d) By drawing a radius as a starting point and using aprotractor, you can construct angles at the centrecorresponding to each sector.
25 This type of graph is extremely helpful to make comparisons This type of graph is extremely helpful to make comparisons. You can see at a glance which slice of the pie represents the biggest/smallest group.The values in a pie-chart are represented by pieces of a pie. The bigger the value, the greater the angle in the piece of pie.
26 Test Your KnowledgeSelect the correct option to indicate the characteristics of a pie diagram…. A: Bars, classes, width of rectangles and class boundaries. B: Circle, sectors, percentages, degrees C: Bars not touching, vertical/horizontal columns, heights and frequencies.
27 The sales are divided into four quarters The sales are divided into four quarters. Illustrate the quarterly sales in a pie-chart.First quarter sales: = 10Second quarter sales: = 14Third quarter sales: = 20Fourth quarter sales: = 28Total sales: = 72
31 Use line and broken line diagrams to represent data A line graph connects points, representing data items in the plane, by means of line segments.Each point corresponds to an item on the horizontal and on the vertical axis.Items on both axes are evenly spaced, according to a appropriate scale.We join the points to highlight the trend in the variable we are measuring.There are two variables, but they are independent of one another.The vertical height of each point is proportional to the quantity being represented.
32 Do not confuse a broken line graph with a frequency polygon. Example: Other points on the lines joining two points have no meaning in terms of the graph.Do not confuse a broken line graph with a frequency polygon.Example:Consider the gold price over an eight week period at the beginning of 2008:Week12345678Price / ounce in $274279276281285283270The graphical representation is given on the following slide
34 Test Your KnowledgeWhat is the difference between a line or broken line diagram and a frequency polygon?Line graphs join midpoints of grouped classesPolygons join points of specific dataPolygons join midpoints of grouped classes and linegraphs join points representing specific data