Quantitative Business Methods (QBM) - Bahrain Polytechnic

Quantitative Business Methods QBM  Chapter 3 Numerical data summary  A Set of data is often summarised and described by two parameters: a Measure of Centrality a Measure of Spread  Measures of Centrality Mode Most commonly occurring value. M...

Quantitative Business Methods QBM  Chapter 3 Numerical data summary  A Set of data is often summarised and described by two parameters: a Measure of Centrality a Measure of Spread  Measures of Centrality Mode Most commonly occurring value. Median Middle value when data is ordered Mean Arithmetic average 4 Formulae for calculating means x represents the x   x value of the data n f the frequency of that particular value  f x  or x  is shorthand for n 'the mean‘ ∑ is shorthand for 'the sum of'  Data Values These may be: Discrete, i.e. individual values People, cars, etc Continuous, i.e. ranges, not individual values Lengths, weights, etc Single Each value stated separately Grouped Frequencies stated for each value or range Examples Discrete data Number of days during which nine members of staff did not use their cars: 2 6 2 4 1 4 3 1 1 In order: 1 1 1 2 2 3 4 4 6 24  2.6 7 Mode: 1 Median: 2 Mean: 9 Grouped discrete data – similar numbers of idle days for 95 staff cars Idle days No. of cars (f) fx Mode (x) Most common number 0 5 0 of days = 2 1 24 24 2 30 60 Median 3 19 57 Middle no.,48th number 4 10 of days = 2 5 5 Mean 6 2 Total number of days = Totals 95 218 18 = 2.30 Total number of staff 95  Grouped continuous data – Fuel invoices for 34 staff car users were: No. of car Value range (£) Mid-value (x) fx users Modal class 59 but < 60 59.5 2 119.0 (range): = £64 to 60 but < 61 60.5 5 302.5 £65 61 but < 62 61.5 4 246.0 62 but < 63 62.5 6 375.0 Medial class 63 but < 64 63.5 5 (range): = £62 to £ 64 64 but < 65 64.5 7 65 but < 66 65.5 3 Mean: 66 but < 67 66.5 2 total value of Totals 34 2141.0 invoices = total number of car users £2141 = £62.97  Measures of Spread The data must be at least ordinal Range Largest value – smallest value Inter-quartile range The range of the middle half of the ordered data Semi-inter-quartile range Half the value of the inter-quartile range Standard deviation The standard statistical measure of spread Population standard deviation (xσn) (The standard measure of how the data is grouped around the mean) x represents each data value  x  x  2 σ  f the frequency of that n particular value or n is the sample size x is shorthand for the mean σ   f x  x 2 value n σ is the shorthand for the f o r f r e q ue nc y d at a population standard deviation ∑ is shorthand for 'the sum of' √ means 'take the positive 11 Sample standard deviation (xσn-1) Usually only a sample is available for analysis x represents each data value f the frequency of that  x  x  2 s  particular value n1 n is the sample size or x is shorthand for the mean value  f x  x  2 s  σ is the shorthand for the n1 population standard f o r f r e q ue nc y d at a deviation ∑ is shorthand for 'the sum of' √ means 'take the positive square root of' as the Examples Discrete data (same data as used for centrality) Number of days during which nine members of staff did not use their cars: 1 1 1 2 2 3 4 4 6 (in order) Range: 6 – 1 = 5 days Examples Discrete data (same data as used for centrality) Interquartile range: Lower quartile ¼ of (9+1) = 2.5th value = 1 day Upper quartile ¾ of (9+1) = 7.5th value = 4 days 4 – 1 = 3 days Semi-inter-quartile range: ½ of 3 = 1.5 days Standard deviation (from calculator) For this sample only: 1.633 days As estimator of all staff cars: 1.732 days Grouped discrete data – similar numbers of idle days for 95 staff cars Range: 6 – 0 = 6 days Idle days No. of cars (f) Cumulativ (x) e (f) Inter-quartile range: 0 5 5 Lower quartile = ¼(95+1) = 24th = 1 day 1 24 29 Upper quartile = ¾(95+1) = 72nd 2 30 59 = 3 days 3 19 78 Inter-quartile range =2 days 4 10 88 5 5 93 Semi-inter-quartile range 6 2 95 2/2 = 1 day Standard deviation (from calculators) For this sample only = 1.345  Grouped discrete data – similar numbers of idle days for 95 staff cars Idle days No. of cars (f) Cumulativ Range: 6 – 0 = 6 days (x) e (f) 0 5 5 Inter-quartile range: 1 24 29 2 30 59 Lower quartile = 3 19 78 ¼(95+1) = 24th 4 10 88 =1 5 5 93 day 6 2 95 Upper quartile = ¾(95+1) = 72nd =3 days  Grouped continuous data – Fuel invoices for 34 staff car users were: Value range (£) Mid- No. of car fx value (x) users 59 but < 60 59.5 2 119.0 60 but < 61 60.5 5 302.5 61 but < 62 61.5 4 246.0 62 but < 63 62.5 6 375.0 63 but < 64 63.5 5 64 but < 65 64.5 7 65 but < 66 65.5 3 66 but < 67 66.5 2 Totals 34 2141.0  Grouped continuous data completed table: Value range (£) Mid- No. of car fx Inter-quartile range: value (x) users 59 but < 60 59.5 2 119.0 Usually estimated from 60 but < 61 60.5 5 302.5 an ogive (see later) 61 but < 62 61.5 4 246.0 Semi-inter-quartile range: 62 but < 63 62.5 6 375.0 Half inter-quartile range 63 but < 64 63.5 5 317.5 = 2/2 = £1 64 but < 65 64.5 7 451.5 Standard deviation: 65 but < 66 65.5 3 196.5 (from calculator) 66 but < 67 66.5 2 133.0 For this sample only Totals 34 2141.0 £1.929 As population estimator £1.958 Estimation of mode from a histogram Draw crossed Histogram of Car mileage as drawn last week Freq. per 10 mile interval diagonals 12 through 10 protruding 8 section of 6 highest column 4 2 Estimated 0 190 100 110 120 130 140 150 160 170 180 mode: 146 miles Mileages 19  Estimation of Median and Inter-quartile range from a Cumulative Frequency Diagram (Ogive) % Cum. Freq. 100 The Median value is that of 90 the 50th percentile: 80 = 146 miles 70 60 The upper quartile is that of 50 the 75th percentile: = 156 miles 40 30 The lower quartile is at of 20 the 25th percentile: 10 = 133 miles 0 100 120 140 160 180 200 The Inter-quartile range is Mileages the difference between the last two: = 23 miles Other useful information This lecture has covered most aspects of numerical summarisation. One other concept that will be useful to you later is that of 'skewness' which you can find in Section 3.6 of Business Statistics for Non-Mathematicians.  In this lecture we learned How to replace data sets by typical numbers – averages- representing them as an aid to a better understanding That statistics such as mean, median and mode can provide easy to calculate single measure of centrality, but care should be taken when choosing the most appropriate for the type of data The importance of measures of variability of a data set about its centre. That the most appropriate statistics depends on the type of data: Ordinal data- use the inter-quartile range about the median Interval or Ratio data- use standard deviation about the mean How to estimate summary statistics from graphically presented data. Questions