1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 7 th edition. Prepared by Lloyd Jaisingh, Morehead State University

2 Using Statistics 使用統計 Percentiles and Quartiles 百分位數與四分位數 Measures of Central Tendency 集中傾向之衡量 Measures of Variability 變異性之衡量 Grouped Data and the Histogram 群聚數據與直方圖 Skewness and Kurtosis 偏態與峰態 Relations between the Mean and Standard Deviation Methods of Displaying Data Exploratory Data Analysis 探索性資料分析 Using the Computer 使用電腦 Introduction and Descriptive Statistics 1

3 Distinguish between qualitative data and quantitative data. Describe nominal, ordinal, interval, and ratio scales of measurements. Describe the difference between population and sample. Calculate and interpret percentiles and quartiles. Explain measures of central tendency and how to compute them. Create different types of charts that describe data sets. Use Excel templates to compute various measures and create charts. LEARNING OBJECTIVES After studying this chapter, you should be able to:

4 Statistics is a science that helps us make better decisions in business and economics as well as in other fields. Statistics teaches us how to summarize, analyze, and draw meaningful inferences from data that then lead to improve decisions. These decisions that we make help us improve the running, for example, a department, a company, the entire economy, etc. WHAT IS STATISTICS ?

Using Statistics (Two Categories) l Inferential Statistics 推論統計 Predict and forecast values of population parameters Test hypotheses about values of population parameters Make decisions l Descriptive Statistics 敘述統計 Collect Organize Summarize Display Analyze

6 l Qualitative 定性 - Categorical or Nominal: Examples are- Color 顏色 Gender 性別 Nationality 國籍 Quantitative 定量 - Measurable or Countable: Examples are- Temperatures 溫度 Salaries 薪水 Number of points scored on a 100 point exam Types of Data - Two Types (p.4)

7 Scales of Measurement (p.4-5) 衡量尺度定 義特 色 可衡量變數 名目尺度 用以辨識或歸 類個案之數值 (1) 僅名目本身有意義，數字大小無意義。 (2) 是層級最低之尺度，無法轉換為其他尺度。 (3) 僅能命名，不能排序。 性別、職業、 居住區域、身 分證碼、學號 等。 順序尺度 用以表示等級 或順序之數值 (1) 只能指出等級或順序，但無法衡量等級間之差異程度。 (2) 可降低為名目尺度。 (3) 僅能排序，不能加減。 名次、排序、 百分位序等。 區間尺度 用以表示程度 上差異之數值 (1) 具任意原點， 0 不代表「無」。 (2) 可降低為順序尺度及名目尺度。 (3) 僅能加減，不能乘除。 溫度、態度、 滿意度、同意 度、重要性等。 比率尺度 用以衡量實質 上差異之數值 (1) 具絕對原點， 0 代表「無」 。 (2) 可降低為區間尺度順序尺度及名目尺度等。 (3) 能乘除。 所得、銷售額、 考試分數、玉 米收穫量等。 Categorical or nonmertric type Nominal scale ( 名目尺度 ) Ordinal scale ( 順序尺度 ) Analytical or metric type Interval scale ( 區間尺度 ) Ratio scale ( 比率尺度 )

8 A population( 母體 ) consists of the set of all measurements for which the investigator is interested. A sample( 樣本 ) is a subset of the measurements selected from the population. l A census( 普查 ) is a complete enumeration of every item in a population. Samples and Populations( 樣本與母體 ) P.5

9 Sampling( 抽樣 ) from the population is often done randomly( 隨機 ), such that every possible sample of equal size (n) will have an equal chance of being selected. l A sample selected in this way is called a simple random sample or just a random sample. l A random sample allows chance to determine its elements. Simple Random Sample

10 Population (N)Sample (n) Samples and Populations

11 Census( 普查 ) of a population may be: Impossible( 不可能 ) Impractical( 不實際 ) Too costly( 成本高 ) Why Sample?

12 Exercise (p.8, 5min)

13 l Given any set of numerical observations, order them according to magnitude. The P th percentile in the ordered( 已排序 ) set is that value below which lie P% (P percent) of the observations in the set. l The position of the P th percentile is given by (n + 1)P/100, where n is the number of observations in the set. 1-2 Percentiles( 百分位數 ) and Quartiles( 四分位數 )

14 20 A large department store collects data on sales made by each of its salespeople. The number of sales made on a given day by each of 20 salespeople is shown on the next slide. Also, the data has been sorted in magnitude. Example 1-2 (p.9)

15 Example 1-2 (Continued) - Sales and Sorted Sales Sales Sorted Sales

16 l Find the 50th, 80th, and the 90th percentiles of this data set. l To find the 50th percentile, determine the data point in position (n + 1)P/100 = (20 + 1)(50/100) = l Thus, the percentile is located at the 10.5th position. l The 10th observation is 16, and the 11th observation is also 16. l The 50th percentile will lie halfway between the 10th and 11th values and is thus 16. Example 1-2 (Continued) Percentiles

17 l To find the 80th percentile, determine the data point in position (n + 1)P/100 = (20 + 1)(80/100) = l Thus, the percentile is located at the 16.8th position. l The 16th observation is 19, and the 17th observation is also 20. l The 80th percentile is a point lying 0.8 of the way from 19 to 20 and is thus Example 1-2 (Continued) Percentiles

18 l To find the 90th percentile, determine the data point in position (n + 1)P/100 = (20 + 1)(90/100) = l Thus, the percentile is located at the 18.9th position. l The 18th observation is 21, and the 19th observation is also 22. l The 90th percentile is a point lying 0.9 of the way from 21 to 22 and is thus Example 1-2 (Continued) Percentiles Example 1-2

19 Quartiles( 四分位數 ) are the percentage points that break down the ordered data set into quarters. l The first quartile is the 25th percentile. It is the point below which lie 1/4 of the data. l The second quartile is the 50th percentile. It is the point below which lie 1/2 of the data. This is also called the median( 中位數 ). l The third quartile is the 75th percentile. It is the point below which lie 3/4 of the data. Quartiles – Special Percentiles( 特殊百分 位數,p.10 )

20 l The first quartile, Q 1, (25th percentile) is often called the lower quartile( 下四分位數 ). l The second quartile, Q 2, (50 th percentile) is often called median or the middle quartile( 中四分位數 ). l The third quartile, Q 3, (75th percentile) is often called the upper quartile( 上四分位數 ). l The interquartile range( 四分位數間距 ) is the difference between the first and the third quartiles. Quartiles and Interquartile Range

21 SortedSales First Quartile Median Third Quartile (n+1)P/100 (20+1)25/100=5.25 (20+1)50/100=10.5 (20+1)75/100= (.25)(1) = (.5)(0) = (.75)(1) = Quartiles Example 1-3: Finding Quartiles Position (16-16) Basic Stat.xls

22 Example 1-3: Using the Template

23 Example 1-3 (Continued): Using the Template This is the lower part of the same template from the previous slide.

24 Exercise, p.11, 10 min 1-9(Ans ： Q1=9, Q2=11.6, Q3=15.5, 55%=12.32, 85%=16.5) 1-12(Ans ： median=51, Q1=30.5, Q3= IQR=163.75, 45%=42.2) Basic Stat.xls P %= (n+1)P / 100

25 Measures of Variability( 衡量變異 性 ) Range 全距 Interquartile range 四分位間距 Variance 變異數 Standard Deviation 標準差 l Measures of Central Tendency( 衡量集中傾向 ) Median 中位數 Mode 眾數 Mean 平均數 l Other summary measures: 其他 Skewness 偏態 Kurtosis 峰態 Summary Measures: Population Parameters Sample Statistics

26  Median 中位數 â Middle value when sorted in order of magnitude â 50th percentile  Mode 眾數 â Most frequently- occurring value  Mean 平均數 â Average 1-3 Measures of Central Tendency or Location(p.11)

27 SalesSorted Sales Median 50th Percentile (20+1)50/100= (.5)(0) = 16 The median is the middle value of data sorted in order of magnitude. It is the 50 th percentile. Example – Median (Data is used from Example 1-2) See slide # 19 for the template output

:. : : : :. : : : Mode = 16 The mode is the most frequently occurring value. It is the value with the highest frequency. Example - Mode (Data is used from Example 1-2) See slide # 19 for the template output

29 The mean( 平均數 ) of a set of observations is their average - the sum of the observed values divided by the number of observations. Population Mean 母體平均數 Sample Mean 樣本平均數    x N i N 1 Arithmetic Mean or Average x x n i n    1 n

30 x x n i n    Sale s Example – Mean (Data is used from Example 1-2) See slide # 19 for the template output

:. : : : :. : : : Median and Mode = 16 Mean = Example - Mode (Data is used from Example 1-2) See slide # 19 for the template output 每一點代表一個數值

32 Exercise, p.15, 5 min 例 ~ 1-16 (See Textbook p.656) 1-17 (Ans ： mean=18.34, median=19.1)

33 l Range 全距 Difference between maximum and minimum values Interquartile Range 四分位數間距 Difference between third and first quartile (Q 3 - Q 1 ) Variance 變異數 Average * of the squared deviations from the mean l Standard Deviation 標準差 Square root of the variance   Definitions of population variance and sample variance differ slightly. 1-4 Measures of Variability or Dispersion (p.15)

34 Sorted SalesSalesRank First Quartile Third Quartile Q 1 = 13 + (.25)(1) = Q 3 = 18+ (.75)(1) = Minimum Maximum Range Maximum - Minimum = = 18 Interquartile Range Q3 - Q1 = = 5.5 Example - Range and Interquartile Range (Data is used from Example 1-2)

35 ( )                ()x N x N N i N i N x i N Population Variance 母體變異數     s xx n x x n n s s i n i n i n              () Sample Variance 樣本變異數 Variance and Standard Deviation ( )

36 公式證明

Calculation of Sample Variance

38 Example: Sample Variance Using the Template Note: This is just a replication of slide #19.

39 Exercise, p.22, 10 min 標準差之計算 - 例 1- 5, 1- 6 (p.20) 或例 (p.22) 1-19 (Ans. Range=27, , ) 1-20 (Ans. Range=60, , ) 1-21 (Ans. Range=1186, , ) Basic Stat.xls

40 l Dividing data into groups or classes or intervals l Groups should be: Mutually exclusive 群間互斥 Not overlapping - every observation is assigned to only one group Exhaustive 完全分群 Every observation is assigned to a group Equal-width (if possible) 等寬 First or last group may be open-ended 1-5 Group Data and the Histogram 群聚數據與直方圖

41 Table with two columns 兩行 listing: Each and every group or class or interval of values Associated frequency of each group Number of observations assigned to each group Sum of frequencies is number of observations –N for population –n for sample Class midpoint 組中點 is the middle value of a group or class or interval Relative frequency 相對頻率 is the percentage of total observations in each class Sum of relative frequencies = 1 Frequency Distribution 頻率分配

42 xf(x)f(x)/n Spending Class ($)Frequency (number of customers) Relative Frequency 0 to less than to less than to less than to less than to less than to less than xf(x)f(x)/n Spending Class ($)Frequency (number of customers) Relative Frequency 0 to less than to less than to less than to less than to less than to less than Example of relative frequency: 30/184 = Sum of relative frequencies = 1 Example 1-7: Frequency Distribution p.23

43 x F(x) F(x)/n Spending Class ($)Cumulative Frequency Cumulative Relative Frequency 0 to less than to less than to less than to less than to less than to less than x F(x) F(x)/n Spending Class ($)Cumulative Frequency Cumulative Relative Frequency 0 to less than to less than to less than to less than to less than to less than cumulative frequency 累積頻率 The cumulative frequency 累積頻率 of each group is the sum of the frequencies of that and all preceding groups. cumulative frequency 累積頻率 The cumulative frequency 累積頻率 of each group is the sum of the frequencies of that and all preceding groups. Cumulative Frequency Distribution

44 頻率分配圖練習, 10 min 例 1- (p.25), 以 5 為距離 Basic Stat.xls

45 histogram A histogram is a chart made of bars of different heights. 不同高度之條狀圖 Widths and locations of bars correspond to widths and locations of data groupings 寬度與 位置代表群組的資料寬度與位置 Heights of bars correspond to frequencies or relative frequencies of data groupings 高度代 表頻率 Histogram 直方圖

46 Frequency Histogram Histogram Example ： 1-7

47 Relative Frequency Histogram Histogram Example

48 l Skewness –Measure of asymmetry of a frequency distribution Skewed to left 左偏 <0 Symmetric or unskewed 對稱 Skewed to right 右偏 >0 l Kurtosis –Measure of flatness or peakedness of a frequency distribution Platykurtic (relatively flat) Mesokurtic (normal) Leptokurtic (relatively peaked) * 公示如 p Skewness 偏度 and Kurtosis 峰度 p.25

49 Skewed to left Skewness 偏度值 -, 越左偏

50 Skewness Symmetric

51 Skewness Skewed to right 偏度值 +, 越右偏

52 Kurtosis Platykurtic 平扁 - flat distribution 扁度值越小, 越平扁

53 Kurtosis Mesokurtic - not too flat and not too peaked

54 Kurtosis Leptokurtic 尖扁 - peaked distribution 扁度值越大, 越尖突

55 Chebyshev’s Theorem 柴比雪夫定理 Applies to any distribution, regardless of shape 可應用 於任何分配之數據 Places lower limits on the percentages of observations within a given number of standard deviations from the mean Empirical Rule r 經驗法則 Applies only to roughly mound-shaped and symmetric distributions 適用山型與對稱之數據 Specifies approximate percentages of observations within a given number of standard deviations from the mean 1-7 Relations between the Mean and Standard Deviation p.27 ( 重要 )

56 l At least of the elements of any distribution lie within k standard deviations of the mean At least Lie within Standard deviations of the mean Chebyshev’s Theorem

57 l For roughly mound-shaped and symmetric distributions, approximately: Empirical Rule 經驗法則

58 Exercise, p.28, 10 min Exercise Basic Stat.xls

59 Pie Charts 圓餅圖 Categories represented as percentages of total Bar Graphs 直條圖 Heights of rectangles represent group frequencies Frequency Polygons 頻率圖 Height of line represents frequency Ogives 累加頻率圖 Height of line represents cumulative frequency Time Plots 時間圖 Represents values over time 1-8 Methods of Displaying Data

60 Pie Chart

61 Bar Chart Average Revenues Average Expenses Fig Airline Operating Expenses and Revenues Airline AmericanContinentalDeltaNorthwestSouthwestUnitedUSAir

62 Relative Frequency Polygon Ogive Frequency Polygon and Ogive Relative Frequency Sales Cumulative Relative Frequency Sales

63 Time Plot

64 圖形練習, 10 min xls 1-25.xls

65 Stem-and-Leaf Displays 莖葉 Quick-and-dirty listing of all observations 快速瀏覽所有觀測值 Conveys some of the same information as a histogram 將資料轉化 成直方圖 Box Plots 盒形圖 Median Lower and upper quartiles Maximum and minimum Techniques to determine relationships 關係 and trends 趨勢, identify outliers 離群值 and influential 有影響的 observations, and quickly describe 快速描述 or summarize 總結 data sets. 1-9 Exploratory Data Analysis – EDA 探 索性資料分析

(10 ~) (20 ~) (30 ~) (40 ~) (50 ~) 6 02 (60 ~) (10 ~) (20 ~) (30 ~) (40 ~) (50 ~) 6 02 (60 ~) Example 1-8: Stem-and-Leaf Display

67 XX *o Median Q1Q1 Q3Q3 Inner Fence Inner Fence Outer Fence Outer Fence Interquartile Range Smallest data point not below inner fence Largest data point not exceeding inner fence Suspected outlier Outlier Q 1 -3(IQR) Q (IQR)Q (IQR) Q 3 +3(IQR) Elements of a Box Plot Box Plot 盒形圖 離群值 IQR 一半數據在盒內

68 Example: Box Plot

69 Exercise, p.39, 15 min BoxPlot.xls

Using the Computer – The Template Output

71 Using the Computer – Template Output for the Histogram

72 Using the Computer – Template Output for Histograms for Grouped Data

73 Using the Computer – Template Output for Frequency Polygons & the Ogive for Grouped Data

74 Using the Computer – Template Output for Two Frequency Polygons for Grouped Data

75 Using the Computer – Pie Chart Template Output

76 Using the Computer – Bar Chart Template Output

77 Using the Computer – Box Plot Template Output

78 Using the Computer – Box Plot Template to Compare Two Data Sets

79 Using the Computer – Time Plot Template

80 Using the Computer – Time Plot Comparison Template