Quantitative Business Methods (QBM) Lecture Notes PDF

Quantitative Business Methods QBM  Chapter 2 Data Presentation Graphs  In this lecture: Histograms Frequency polygons Stem and leaf plots Cumulative frequency diagrams Also: Barcharts Piecharts Dotplots stem-and-...

Quantitative Business Methods QBM  Chapter 2 Data Presentation Graphs  In this lecture: Histograms Frequency polygons Stem and leaf plots Cumulative frequency diagrams Also: Barcharts Piecharts Dotplots stem-and-leaf plots boxplots, etc. (All in Business studies for Mathematicians)  Histogram A pictorial method for representing frequency data. Similar to a barchart but: Data must be measurable, e.g. lengths rather than colours The area of a block is proportional to the frequency. For unequal intervals frequency density is plotted.  Frequency distribution table A table of grouped data Clearer picture than individual data values Usually presented in the form of a histogram Also: barcharts, piecharts, dotplots, stem-and-leaf plots, boxplots, are used for frequency data.  First: Construct a frequency distribution table (from raw data) Steps 1. Find the range 2. Decide class intervals; all same at this stage. 3. Construct frequency distribution table. 4. If data scarce at extremes join classes. 5. If central frequencies very high split classes. 6. If class intervals not all same calculate frequency densities. Example: (Steps 1 and 2) Mileages recorded for a sample of hired vehicles during a given week yield the following data: 138 164 150 132 144 125 149 157 146 158 140 109 136 148 152 144 168 126 138 186 163 109 154 165 146 183 105 108 135 153 140 135 161 145 135 142 150 156 145 128 Min = 105, Max. = 186 Range = 81 miles Nine intervals of 10 miles width seems sensible Produce a frequency distribution table (Step 3) 138 164 150 132 144 Class Interval Frequency 125 149 157 146 158 100 & < 110 4 140 109 136 148 152 110 & < 120 0 120 & < 130 3 144 168 126 138 186 130 & < 140 7 163 109 154 165 146 140 & < 150 11 183 105 108 135 153 150 & < 160 8 140 135 161 145 135 160 & < 170 5 170 & < 180 0 142 150 156 145 128 180 & < 190 2 Produce frequency density table because data is scarce at the extremes Steps 4-6 Frequency Density= frequency / class width Class Interval Frequency (per 10 miles) 100 & < 110 4 Class Interval Frequency Freq. 110 & < 120 0 dens. 120 & < 130 3 130 & < 140 7 100 & < 110 4 140 & < 150 11 110 & < 120 0 2.0 150 & < 160 8 120 & < 130 3 3.0 160 & < 170 5 130 & < 140 7 7.0 170 & < 180 0 180 & < 190 2 140 & < 150 11 11.0 Total 40 150 & < 160 8 8.0 10 miles = 1 class width 160 & < 170 5 5.0 170 & < 180 0 1.0 180 & < 190 2 Total 40.0 Draw the histogram in your handout 1. For given frequency table, close any open intervals with sensible limits. 2. Label each axis carefully; frequency or frequency density usually shown vertically. 3. Horizontal axis shows value at interval centre for discrete data, at interval limits for continuous data. 4. N.B. Areas proportional to frequency so if width is doubled then halve height for same frequency 5. Horizontal axis shows value at interval centre for discrete data, at interval limits for  Histogram Freq. per 10 mile interval 12 10 8 6 4 2 0 80 100 120 140 160 180 200 Mileages  Comparing the presentation of frequency and frequency density Histogram of Mileages Histogram of Mileages 10 10 10 mile interval Frequencyper Frequency 5 5 0 0 100 110 120 130 140 150 160 170 180 190 100 110 120 130 140 150 160 170 180 190 Mileages Mileages A frequency polygon may be added by joining the mid-points of the column tops Freq. per 10 mile interval 12 10 8 6 4 2 0 80 100 120 140 160 180 200 Mileages Stem and leaf plots: Same 'shape' as the histogram. Generally shown horizontally. Stem: most significant figure Leaves: next significant shown in order. All the original information kept as the numbers themselves are retained in the diagram. Example: Same data as for the previous histogram Freq. Stem Leaf The 'stem' indicates, 4 10 | 5789 in this particular 0 11 | example, the first two 3 12 | 568 digits, (hundreds and 7 13 | 2555688 11 14 | 00244556689 tens), 8 15 | 00234678 5 16 | 13458 the 'leaf' the last one, 0 17 | 2 18 | 36 (the units).  Cumulative frequency diagrams ( Ogives, Cumulative frequency polygons) A graphical method of representing the accumulated frequencies below and including a particular value, i.e. a running total. Percentage cumulative frequencies are often calculated. Method is used for estimating Median and Quartile values. Cumulative frequency diagrams ( Ogives, Cumulative frequency polygons) Interquartile or semi-interquartile range of the data used as measure of spread. Also used to estimate the percentage of the data above or below a certain value. Method 1. Construct a frequency table as before. 2. Construct a cumulative frequency table. 3. Calculate the cumulative percentages. 4. Plot the cumulative percentage on the vertical axis against the end of the interval and join up the points with straight lines. Plot this data in your handout. Cumulative frequency distribution (fill in table in handout) Interval Less Than Freq. Cum.Freq. % Cum. Freq. 100 0 0 0.0 120 4 4 10.0 130 3 7 17.5 140 7 14 35.0 150 11 160 8 170 5 190 2 40 100.0 40 Cumulative frequency distribution (completed table) Interval less than Frequency Cumulative freq. % Cum. Freq. 100 0 0 0.0 120 4 4 10.0 130 3 7 17.5 140 7 14 35.0 150 11 25 62.5 160 8 33 82.5 170 5 38 95.0 190 2 40 100.0 Total 40  Cumulative frequency diagram % Cum. Freq. 100 90 80 70 60 50 40 30 20 10 0 100 120 140 160 180 200 Mileages  Other graphical presentation In this lecture we have concentrated on the graphical presentation of continuous data in the form of histograms and cumulative frequency diagrams. Much data that you will come across is only nominal and ordinal so you will need to be able to construct and interpret bar charts and pie charts. These are considered in detail in Business Statistics for Non- Mathematicians in Section 2.4. There is also more practice in constructing histograms and cumulative frequency Next lecture: Today’s diagrams will be needed during the next lecture: The histogram will be used for estimating the modal mileage. The cumulative frequency diagram will be used for estimating median and quartile mileages Summary In this Chapter we have looked at Various ways of presenting data in graphical form. The choice of presentation depends mainly on the scale of measurement of the data. Scale of measurement:  Nominal: Bar charts, dot plot, pie chart  Ordinal: Bar charts, dot plot, pie charts, box and whisker plot  Interval or Ratio: Box and whisker plot, stem and leaf plot, histogram, frequency polygon, cumulative frequency polygon Questions