Graphical Data Summaries and Presentations
Learn about various graphical presentations of data including stem-and-leaf displays, histograms, and bar charts to effectively summarize examination scores for 80 students. Understand how these visualizations help in analyzing and interpreting data.
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Summarizing Data Graphical Presentations
Table 1: Examination scores for 80 students 72 38 43 81 79 71 65 59 90 83 39 42 58 56 72 63 49 81 56 60 83 89 60 52 62 32 28 39 49 48 65 72 81 58 45 52 63 73 69 75 91 49 67 76 72 60 40 58 52 68 54 52 58 77 88 70 61 39 74 68 29 36 49 62 31 73 40 38 59 60 75 93 53 57 61 65 70 79 37 46
Stem-and-leaf display 2 3 4 5 6 7 8 9 8 9 1 8 8 2 9 9 9 6 7 3 5 0 9 2 9 8 0 9 6 9 2 8 6 4 2 2 8 9 9 2 3 7 8 6 8 8 0 3 0 5 0 9 2 1 7 1 8 5 0 5 3 2 3 9 1 3 7 5 5 0 4 6 2 0 9 2 2 2 1 1 1 3 9 8 3 3 0 1
Stem-and-leaf display 2 3 4 5 6 7 8 9 8 9 1 2 6 7 8 8 9 9 9 0 2 3 5 6 8 9 9 9 9 0 2 2 2 2 3 4 6 6 7 8 8 8 8 9 9 0 0 0 0 1 1 2 2 3 3 5 5 5 7 8 8 9 0 1 2 2 2 2 3 3 4 5 5 6 7 9 9 0 1 1 1 3 3 8 9 0 1 3
Stem-and-leaf display Break each number into its tens and units digits. Tally together values which share the tens digit. The ten digits will then be aligned vertically with the units digits displayed to the side.
Histogram The most common form of graphical presentation of frequency Constructed by representing the measurements that are grouped on a horizontal scale and the frequencies on a vertical scale.
Histogram Examination scores for 80 students 16 14 12 10 8 6 4 Std. Dev = 16.04 Mean = 60.5 2 N = 80.00 0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0 SCORES
Bar chart Similar to histograms, the heights of the bars represent the frequencies but there is no pretense of having a continuous horizontal scale. Useful when groups are categorical.
Bar chart Examination scores for 80 students 20 10 Count 0 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 CLASSES
Line Graph Examination scores for 80 students 20 10 Count 0 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 CLASSES
Pie chart Percentages of categories. A circle is divided into sectors (pie-shaped pieces) which are proportional in size to the corresponding percentages Avoid 3-D pie chart!!
Pie chart Examination scores for 80 students 91-100 21-30 2.5% 2.5% 81-90 31-40 10.0% 13.8% 71-80 41-50 17.5% 11.2% 61-70 51-60 18.8% 23.7%
Pie chart Financial Assistance for Students of HKBU Others .7% Scholarships 1.8% Loans 1.1% Government Grant Government Loan 51.6% 44.8%
Pictogram Pictorial presentations of data. Often seen in newspapers, magazines, and reports of various sorts. For non-statistical-oriented readers.
Summarizing Data Tabular Presentations
Tabular vs Graphical When we deal with a large data set, a good overall picture can be a graphical presentation such as bar chart (for nominal and ordinal data) or histogram (for interval and ratio data). However, little statistical analysis can be done if we have only graphical presentations. Another good overall picture which allows deeper statistical analysis is a tabular presentation.
Frequency distribution of categorical data Table 2: Responses of young boys to removal of toy Response Cry Express anger Withdraw Play with another toy Total Frequency 25 15 5 5 50 Organizing data? Isn t this table the original raw data?
Table 1: Examination scores for 80 students 72 38 43 81 79 71 65 59 90 83 39 42 58 56 72 63 49 81 56 60 83 89 60 52 62 32 28 39 49 48 65 72 81 58 45 52 63 73 69 75 91 49 67 76 72 60 40 58 52 68 54 52 58 77 88 70 61 39 74 68 29 36 49 62 31 73 40 38 59 60 75 93 53 57 61 65 70 79 37 46
Table 3: Examination grades for 80 students B D C A B B B C A A D C C C B B C A C B A A B C B D D D C C B B A C C C B B B B A C B B B B D C C B C C C B A B B D B B D D C B D B D D C C B A C C B B B B D C
Frequency distribution for examination grades Grade A B C D Total Frequency 10 29 28 13 80
Cumulative frequency distribution for ordinal data Grade A B C D Cumulative Frequency 10 39 67 80
(Cumulative) Percentage distribution Grade A B C D Total Percentage 12.5 36.3 35.0 16.3 100 Cumulative percentage 12.5 48.8 83.8 100 The decimal points are at the same vertical position.
Comparing distributions Table 4: Response to removal of toy by gender of child Response Cry Express anger Withdraw Play with another toy 5 Total Male 25 15 5 Female 56 6 8 30 100 50 More girls withdraw?
Comparing distributions Percentage distribution Response Cry Express anger Withdraw Play with another toy 10% Total Male 50% 30% 10% Female 56% 6% 8% 30% 100% 100%
Bar chart 60 50 40 30 20 10 FEMALE Value 0 MALE Cry Express anger Withdraw Another toys RESPONSE
Comparing distributions Making comparisons between distributions is a procedure often used. If the total numbers of cases are equal, the frequency distributions can be used to make comparisons In general, we use percentage distributions to make comparison.
Distribution for ratio data Ratio data are sometimes spread over a wide range, making the resultant frequency (percentage) distribution long and difficult to read.
Examination scores for 80 students 72 38 43 81 79 71 65 59 90 83 39 42 58 56 72 63 49 81 56 60 83 89 60 52 62 32 28 39 49 48 65 72 81 58 45 52 63 73 69 75 91 49 67 76 72 60 40 58 52 68 54 52 58 77 88 70 61 39 74 68 29 36 49 62 31 73 40 38 59 60 75 93 53 57 61 65 70 79 37 46
Frequency/Percentage Distributions
Grouped frequency distribution In order to clarify our presentation, we might construct a grouped frequency distribution. Condense the separate scores into a number of smaller categories or groups, each containing more than one score value.
Grouped distributions Grouped frequency/percentage distributions present raw (unprocessed) data in a more readily usable form. The price for this is the loss of some information. Worthwhile.
Grouped distributions Construction of a frequency/percentage Choosing the classes (intervals or categories) Counting the frequency/calculating the percentages of items in each class Purely mechanical
Grouped distributions Choosing a suitable classification deciding how many classes, from where to where each class should go. Both are arbitrary General rules number of classes usually between 6 and 15; exact number depends largely on how many observations there are; each item must go into one and only one class; whenever possible, each class is of the same length.
Graphical presentation For grouped interval/ratio data, histogram, instead of bar chart, is usually used.
Graphical presentation 20 10 Std. Dev = 16.04 Mean = 60 N = 80.00 0 25 35 45 55 65 75 85 95 SCORES
