Advanced Mathematics for Business Topic 1: Introductory Management Statistics © NCC Education Limited Title of Topic Topic 1 - 1.2 Role of Mathematics in Business Senior managers will use mathematics to assist when making business decisions, for example: • How many goods to produce • What prices to charge • How many employees are required • Whether to invest in new capital assets • Whether to develop a new range of products • Whether to fund a new marketing campaign Title of Topic Topic 1 - 1.3 Scope and Coverage This topic will cover: • Data types • A revision of summary statistics • Index numbers Title of Topic Topic 1 - 1.4 Learning Objectives By the end of this topic students will be able to: • recognise nominal, ordinal, interval and ratio data types • recognise and use mode, median, mean, range, standard deviation and coefficient of variation • calculate Laspeyres and Paasche index numbers • use index numbers to calculate percentage changes and to deflate series Title of Topic Topic 1 - 1.5 Data Types (Level of Measurement) • Nominal Low • Ordinal Information Content • Interval • Ratio High Title of Topic Topic 1 - 1.6 Nominal Data 名目数据 (Named) • Data which can only be categorised. • The data can be counted but not ordered, measured or expressed as a ratio. • Examples include: • Gender (male or female) • Job role (administration, production, accounting, sales) • Product type (fruit cake, chocolate cake, cream cake) • Summary statistics: • Mode 众数 Title of Topic Topic 1 - 1.7 Count • Parts in stock 3 10 9 6 8 7 2 11 6 7 5 12 7 5 8 5 7 8 6 5 9 4 6 10 1 8 3 6 11 6 2 10 7 9 7 4 8 8 4 4 9 5 7 9 3 Part number 7 • Return 6 8 5 9 4 3 10 2 11 1 12 Total for Frequency 7 next point 6 6 5 5 4 3 3 2 2 1 1 45 • Mode - The number or category that occurs the most Title of Topic Topic 1 - 1.8 Ordinal 序数 (Ordered) • Data which can be ranked and counted but can not be measured or expressed as a ratio. • Examples include: • Employee grades (A, B, C …) • Voting preferences (best candidate, second best …) • Product preferences • Summary statistics: • Mode, median 中位数 Title of Topic Topic 1 - 1.9 Order • Score on rating scale 3 10 9 6 8 7 2 11 6 7 5 12 7 5 8 5 7 8 • Median 6 5 9 4 6 10 1 8 3 6 11 6 2 10 7 9 7 4 8 8 4 4 9 5 7 9 3 Score 1 2 3 4 5 6 7 8 9 10 11 12 Frequency 1 2 3 4 5 6 7 6 5 3 2 1 Cumulative frequency 1 3 6 10 15 21 28 34 39 42 44 45 - Value of middle item of ordered data, (N +1)/2 term Title of Topic Topic 1 - 1.10 Interval Data 等距数据 • Data which can be placed along a scale, ranked, measured and counted but cannot be expressed as a ratio. • Examples: – Centigrade temperature scale (10C, 15C, 20C) – IQ score (100, 115, 130) • Summary statistics: – Mode, median, mean – Range, standard deviation Title of Topic Topic 1 - 1.11 Measure • Temperature C 3 10 9 6 8 7 2 11 6 7 5 12 7 5 8 5 7 8 6 5 9 4 6 10 1 8 3 6 11 6 2 10 7 9 7 4 8 8 4 4 9 5 7 9 3 Temperature (x) 1 2 3 4 5 6 7 8 9 10 11 12 Total • Mean - total value of all data / number of data Frequency (f) 1 2 3 4 5 6 7 6 5 3 2 1 45 fx 1 4 9 16 25 36 49 48 45 30 22 12 297 = mean = 297/45 = 6.6C Title of Topic Topic 1 - 1.12 Standard Deviation Title of Topic Topic 1 - 1.13 Measure Temperature Frequency (x) (f) 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 6 9 5 10 3 11 2 12 1 Total 45 fx 1 4 9 16 25 36 49 48 45 30 22 12 297 x- -5.6 -4.6 -3.6 -2.6 -1.6 -0.6 0.4 1.4 2.4 3.4 4.4 5.4 (x - )2 31.36 21.16 12.96 6.76 2.56 0.36 0.16 1.96 5.76 11.56 19.36 29.16 f(x - ) 2 31.36 42.32 38.88 27.04 12.80 2.16 1.12 11.76 28.80 34.68 38.72 29.16 298.8 Title of Topic Topic 1 - 1.14 Standard deviation Ratio 比率数据 (Scale with Natural Zero) Title of Topic Topic 1 - 1.15 • Data which can be placed along a scale, ranked, measured, counted and expressed as a ratio. • Examples – Kelvin temperature scale – Time spent in queue (10, 20, 30 minutes) – Differences on interval scales • Summary statistics – Mode, median, mean – Range, standard deviation, coefficient of variation 变动系数, skewness Title of Topic Topic 1 - 1.16 Ratio Queuing time Frequency (x) (f) 1 1 2 2 etc. etc. 11 2 12 1 total 45 fx 1 4 etc. 22 12 297 x- -5.6 -4.6 etc. 4.4 5.4 (x - )2 31.36 21.16 etc. 19.36 29.16 f(x - ) 2 31.36 42.32 etc. 38.72 29.16 298.8 Data types (Discrete 离散 & Continuous 连续) Title of Topic Topic 1 - 1.17 • Discrete data – Can take a countable number of values • Number of products built (1,237,502) • Number of questions correct in test (4) • Continuous – Can take an uncountable number of values • Length of fabric cut (45.9847248738…metres) • Time in queue (23 minutes 5.2084792…seconds) Title of Topic Topic 1 - 1.18 Recap and Review Summary. 四个数据类型(Nominal, Ordinal, Interval and Ratio )、五个关键公式 (平均、众数、中位数、标准差、变动 系数)、两种数据(离散和连续) Any questions? Title of Topic Topic 1 - 1.19 Index numbers 指数数据 Title of Topic Topic 1 - 1.20 Simple Price Index - 1 Consider chocolate bar, set 2014 as base year Title of Topic Topic 1 - 1.21 Simple Price Index - 2 • Compare the prices of snack products over time Title of Topic Topic 1 - 1.22 Year Chocolate Bar Price Index Sandwich Price Index Bag of Crisps Price Index Jan 2014 0.47 100.00 1.85 100.00 0.60 100.00 Jan 2015 0.52 110.64 1.92 103.78 0.61 101.67 Jan 2016 0.56 119.15 2.00 108.11 0.63 105.00 Title of Topic Topic 1 - 1.23 Percentage Change Between Index Numbers year chocolate bar index annual % sandwich price index bag of crisps price index Jan 2014 100.00 - 100.00 - 100.00 - Jan 2015 110.64 10.64 103.78 3.78 101.67 1.67 Jan 2016 119.15 7.69 108.11 ? 105.00 ? Title of Topic Topic 1 - 1.24 Rebasing Price Relatives • Reasons for changing base period; - Mature series => large indices, relevance of base period - Compare different indices which have used different base years Title of Topic Topic 1 - 1.25 Rebasing Price Relatives Title of Topic Topic 1 - 1.26 Simple Aggregate Indices Title of Topic Topic 1 - 1.27 Laspeyres (Base Weighted) Price Index Title of Topic Topic 1 - 1.28 Example Title of Topic Topic 1 - 1.29 Paasche (Current Weighted) Price Index Title of Topic Topic 1 - 1.30 Example Title of Topic Topic 1 - 1.31 Weighted Index Numbers Weighted Index Numbers 加权指 数 Title of Topic Topic 1 - 1.32 Eggs Flour Fat Sugar Lemons Gas w 18 16 5 14 2 45 100 R 151.00 225.70 94.60 405.00 256.40 319.40 2718.00 3611.20 473.00 5670.00 512.80 14373.00 27358.00 Title of Topic Topic 1 - 1.33 Deflating Data - 1 Title of Topic Topic 1 - 1.34 Deflating Data - 2 Title of Topic Topic 1 - 1.35 Summary At the end of this presentation, you should be able to: • Recognise nominal, ordinal, interval and ratio data types • Recognise and use mode, median, mean, range, standard deviation and coefficient of variation • Calculate Laspeyres and Paasche index numbers • Use index numbers to calculate percentage changes and to deflate series Title of Topic Topic 1 - 1.36 References • ONS www.statistics.gov.uk/statbase/product.asp?vlnk=8 68, accessed 20 December 2015 Topic 1 – Introductory Management Statistics Any Questions?