blacksacademy.net: Further presentation and analysis of data

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Analysis of data SLIDE 1 Consolidation - cumulative frequency and the box plot The table shows information about the distance in metres of the houses of 100 children attending a primary school. Distance / m Frequency Interval Cumulative frequency 6 6 11 16 28 21 11 5 2 Complete the table. SLIDE 2 Distances of houses of children from a primary school Distance / m Frequency Interval Cumulative frequency 6 6 11 17 16 33 28 61 21 82 11 93 5 98 2 100 SLIDE 3 Interval / m Cumulative frequency Interval / m Cumulative frequency 6 82 17 93 33 98 61 100 Plot the distance against the cumulative frequency. SLIDE 4 Find Q1, Q2 and Q3. SLIDE 5 SLIDE 6 Distances of houses of children from a primary school distance / m Lowest 45, 60 Q1 265 Q2 362 Q3 424 Highest 655, 685, 710, 735 Revision The lower fence is either or the least data value, if this value lies within the inter-quartile range. The upper fence is either or the greatest data value. An outlier is any data value not lying within the lower and upper fence. ? Find the lower and upper fences for the data. ? Are any of the data values outliers? SLIDE 7 Distances of houses of children from a primary school distance / m Lowest 45, 60 Q1 265 Q2 362 Q3 424 Highest 655, 685, 710, 735 SLIDE 8 Distances of houses of children from a primary school distance / m Lower fence 26.5 Q1 265 Q2 362 Q3 424 Upper fence 662.5 Outliers 685, 710, 735 Draw a box plot representing the data SLIDE 9 Distances of houses of children from a primary school distance / m Lower fence 26.5 Q1 265 Q2 362 Q3 424 Upper fence 238.5 Outliers 685, 710, 735 SLIDE 10 Describe the skew of the data. Describe what this skew means in words. SLIDE 11 Distances of houses of children from a primary school The data exhibits negative skew. The tail of the distribution is elongated towards the smaller values of the distance. The spread (values) of distances below the median is larger than the spread (variance) of distances above the median. SLIDE 12 Distances of houses of children from a primary school (2) A box plot for a second primary school is shown above. Complete the table. distance / cm Lower fence 110 Q1 610 Q2 790 Q3 1070 690 Upper fence 1760 Outliers 1800 ? Describe the skew of the data. ? What is the least distance of a child�s house from this primary school? Justify your answer. SLIDE 13 Distances of houses of children from a primary school (2) distance / cm Lower fence 110 Q1 610 Q2 790 Q3 1070 460 Upper fence 1760 Outliers 1800 The data exhibits a positive skew. The interquartile range is 460 m. Since , the lowest fence is the least data value. So, the child living closest to the school is represented by the lowest fence, and lives 110 m away. SLIDE 14 Distances of houses of children from a primary school (1) Distances of houses of children from a primary school (2) One of the schools is a small primary school in a densely populated city. The other school is a similarly sized primary school in a small market town. Which school is which? Justify your answer. SLIDE 15 Distances of houses of children from a primary school (1) Distances of houses of children from a primary school (2) The scale of both box plots differs, and the median value for school (2) is 790 m in contrast to the median value for school (1) of 362 m, almost twice the distance. Primary school (1) is likely to be the school from the densely populated city area, because more people live within a given area, and hence children are likely to live closer to the school they attend. SLIDE 16 Continuous data ? What is the distinction between continuous and discrete data? Given an example illustrating the difference. ? What is the difference between a bar chart and a histogram, and when do you use the one rather than the other? SLIDE 17 ? Data that is discrete can be counted by integers. Continuous data is data that can be measured by real numbers. Number of people at football matches ? discrete. Height of mountains ? continuous. ? In a bar chart the columns are separated by gaps. In a histogram, there are no gaps, and measurements apart at the boundaries of the columns. Discrete data ? Bar chart. Continuous data ? Histogram. SLIDE 18 Continuous data and frequency tables ? ? Interval / m Frequency Interval / m Frequency 6 6 11 11 � � � � ? The above shows the beginning of two frequency tables presenting data. There is only one difference between the two tables � what is it? ? State the width of the intervals used in both tables. SLIDE 19 SOLUTION AND EXPLANATION Interval width in a frequency table ? ? Interval / m Interval width / m f Interval / m Interval width / m f 10 6 9.5 6 10 11 10 11 � � � � ? The two tables differ in the way the interval is presented. In the first the boundaries are exact; in the second, the boundaries are given to the nearest integer value. ? The interval width in the first table is , but in the second, because of the way the interval has been written, the interval width varies. stated interval actual interval interval width For example, a value of 9.8 would not fall within the interval , because 9.8 rounds up to 10. It is a favourite trick of examiners to present continuous data using discrete boundaries. The student is expected to interpret the intervals correctly. SLIDE 20 ? ? Interval Interval width f Interval Actual interval Interval width f 3 3 7 7 20 20 Complete the tables. What do you observe about the interval width in both cases, as we proceed down the frequency table? Why might a frequency table be presented in this way? SLIDE 21 SOLUTION AND EXPLANATION Variable class width Interval Interval width f Interval Actual interval Interval width f 15 3 14.5 3 10 7 10.0 7 5 20 5.0 20 The interval widths in both cases are not the same value. A variable interval width is used. Data is presented in this way because there are more data values around the mid-value (mean or median) than there are in the tails. Because data tends to cluster in the middle region, intervals of different width may be presented. Note. The width of an interval is also called the class width. SLIDE 22 Mass of individual strawberries Interval / g f 4 9 22 38 17 10 A student draws the following histogram. Why is the presentation of the data in this way misleading? SLIDE 23 Mass of individual strawberries Interval / g f 4 9 22 38 17 10 This histogram is misleading because it creates the impression that there were more seeds in the tails of the distribution than really was the case. The areas of the intervals representing the tails are far too large. SLIDE 24 Representation of continuous data in a histogram When intervals are of variable width frequency is proportional to the area of the rectangle SLIDE 25 Mass of individual strawberries Interval / g f 4 9 22 38 17 10 ? In this histogram to what is the frequency proportional? ? How must the diagram be changed to rectify the problem that it is misleading? SLIDE 26 SOLUTION AND EXPLANATION Mass of individual strawberries Interval / g f 4 9 22 38 17 10 ? In the above diagram, frequency is proportional to the height of the rectangle. ? This is misleading, the frequency must be proportional to the area of the rectangle. SLIDE 27 Frequency density When intervals are of variable width frequency is proportional to the area of the rectangle Mass of individual strawberries Interval / g f Class width Frequency density 4 15 0.27 9 22 38 17 10 Complete the above table. In the first row SLIDE 28 SOLUTION AND EXPLANATION Interval / g f Class width Frequency density 4 15 0.27 9 10 0.90 22 5 4.4 38 5 7.6 17 10 1.7 10 25 0.4 Draw a histogram to represent this data. SLIDE 29 SOLUTION AND EXPLANATION Interval / g f Class width Frequency density 4 15 0.27 9 10 0.90 22 5 4.4 38 5 7.6 17 10 1.7 10 25 0.4 SLIDE 30 Revision � skew SLIDE 31 The table shows the ages of footballers playing in a professional league. Age / years Frequency Class width Frequency density 15 38 86 152 72 40 6 Complete the table and draw a histogram. Describe the skew of the data. SLIDE 32 Age / years Frequency Class width Frequency density 15 4 3.75 38 2 19 86 2 43 152 2 76 72 4 18 40 10 4 6 6 1 The skew of the data is positive. SLIDE 33 Estimation Age / years Frequency 15 38 86 152 72 40 6 Ages of footballers playing in a professional league Above is a grouped frequency table for the age of footballers in a football league. In the table, the original data of the age of the footballers has been grouped into classes. The first class is . So, the table does not provide exact information about the age of those footballers � all we know is that 15 of them belong to that class. By assuming that the actual ages, when put in rank order, are evenly spaced out in any given class, we can estimate the age of, say, the 10th value in rank order. The method that assumes that the age increases linearly with rank is called linear interpolation. SLIDE 34 Age / years Frequency Cum f 15 15 38 53 86 139 152 291 72 362 40 403 6 409 By linear interpolation to estimate the median age of the footballers to the nearest month. There are 409 footballers in the table. The median value is the 205th footballer in rank order by age. This footballer belongs to the class . Assume that the 139th value is 24 and the 291th value is 26. Then, by using a linear relationship (straight-line), we find by similar triangles. SLIDE 35 A survey was made of the hourly wage of electricians. Hourly wage / � Class width Frequency density Frequency Complete the table. ? Find the total number of electricians participating in the survey. ? Estimate the mean and median. SLIDE 36 Hourly wage / � x Class width Frequency density f xf 9 8 2.5 20 180 19 2 7.5 30 570 22 4 6.0 24 528 37 26 0.5 13 481 Totals 87 1759 The total number is 87. The mean is . The median is the 44th value. It lies in the class . SLIDE 37 80 observations were made of a variable x. Each observation was made to the nearest whole number. x Frequency 30 18 32 A histogram was drawn so that the rectangle representing the class had a width of 2 cm and a height of 5 cm. What was the width and height of the rectangle representing the class? SLIDE 38 � error in interval of last corrected � does not affect solution � CHECK THE SLIDE 80 observations were made of a variable x. Each observation was made to the nearest whole number. x Frequency 30 18 32 A histogram was drawn so that the rectangle representing the class had a width of 2 cm and a height of 5 cm. What was the width and height of the rectangle representing the class? Solution The width of the class is not but . The width of the class is . So the width of the rectangle representing the class is half the width of the rectangle representing the ; this is 1 cm. The area of the class is . Since, frequency area, The class has a frequency of 18 so requires an area of . The area of the class is . So, the height of this class is . Thus, the rectangle for class, has width 1 cm, height 6 cm.

Further presentation and analysis of data

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