All the men living in Turkey can be a population. The average height of these men can be a population parameter

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1 CHAPTER 1: WHY STUDY STATISTICS? Why Study Statistics? Population is a large (or in nite) set of elements that are in the interest of a research question. A parameter is a speci c characteristic of a population All the men living in Turkey can be a population. The average height of these men can be a population parameter Sample is a subset of population that we use to withdraw conclusions or predictions on the parameters of the population (for inferences to be valid, sampling should be random). Statistics is a characteristic of the sample Instead of measuring the height of every man in Turkey, we can randomly select 5000 men from di erent locations of the country. This would be our sample. Then we can nd the average height of these people to estimate average height of the men in Turkey. This would be our sample statistics 1

2 Types of Statistics Inferential Statistics: This is what explained above; i.e. using sample data to make estimation and hypothesis testing (the tools that helps us to make statements and decisions under uncertainty, incomplete information) Descriptive Statistics: Graphical and numerical procedures that are used to present and summarize data. We can use descriptive statistics on either population, or sample data Chapter Summary Terms reviewed in this chapter: Population (Populasyon) Parameter (Parametre) Sample (Örneklem) Inferential Statistics (Ç kar msal Istatistik) Estimation (Tahmin) Descriptive Statistics (Betimleyici Istatistik) 2

3 CHAPTER 2: USING GRAPHS TO DESCRIBE DATA Data, Variable, and Constant Data are usually just a set of numbers representing the same kind of thing, such as body weight. That "thing" is called a variable (it is variable because the numbers vary from subject to subject). If the numbers are the same, the thing is called a constant Classi cation of Variables Categorical (sometimes called Nominal) or Numerical Categorical: (Yes or No), (Like, Dislike or Indi erent),... Numerical: (Discrete: Outcome of a dice,...), (Continuous: Height, time,...) Qualitative or Quantitative Qualitative: These variables are measured on an ordinal, interval, or ratio scale to describe variables. Numerical identi cation is only given to make variables categorized (Yes and No can be labeled as 0 and1). Ordered data indicate the rank of ordering items as well (Like, Dislike and Indi erent can be labelled as 2, 1,0). This thpe of data can be either categorical or numerical Quantitative: They are measured on a nominal scale. Hence, numeric values matter 3

4 Independent or Dependent Independent: A variable that stands alone and isn t changed by the other variables (ex. someone s age) Dependent: A variable that is explained by independent variables 4

5 Tables And Graphs to Describe Categorical Variables The Frequency Distribution Table reveals the number of occurrence (frequency) of each possible outcome A probability distribution is a frequency distribution with each frequency divided by the total number of observations Bar Chart, Pie Chart and Pareto Diagram are the graphics that present the same information with the Frequency Distribution Table Example: Hospital Patients by Unit Frequency Distribution Table Bar Chart Pie Chart Hospital Unit Number of Patients Cardiac Care 1,052 Emergency 2,245 Intensive Care 340 Maternity 552 Surgery 4,630 Number of patients per year Cardiac Care Hospital Patients by Unit Emergency Intensive Care Maternity Surgery Surgery 53% Hospital Patients by Unit Cardiac Care 12% Emergency 25% Intensive Care 4% Maternity 6% 5

6 Pareto Diagram: It is a special Bar Chart. But unlike Bar and Pie Charts, Pareto diagram presents the information in an order (descending or ascending), and the cumulative total is represented by the line Ex: 400 defective items are examined for cause of defect Frequency Distribution Table Source of Manufacturing Error Bad Weld Poor Alignment Missing Part Paint Flaw Electrical Short Cracked case Total Pareto Diagram Number of defects Source of Manufacturing Error Poor Alignment Paint Flaw Bad Weld Missing Part Cracked case Electrical Short Total Arranging Data Number of defects % of Total Defects % Pareto Diagram: Cause of Manufacturing Defect 60% 100% 90% % of defects in each category (bar graph) 50% 40% 30% 20% 10% 0% Poor Alignment Paint Flaw Bad Weld Missing Part Cracked case Electrical Short 80% 70% 60% 50% 40% 30% 20% 10% 0% cumulative %(line graph) 6

7 Tables And Graphs to Describe Numerical Variables We have frequency distribution just like the case with categorical variables. However, since this time the data is not categorized into groups, it is better to form arti cial groups instead of revealing frequency of each data point Ex: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature: 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27 The Ordered Data 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 The Frequency Distribution Table Interval Frequency Relative Frequency Percentage more than 10 but less than more than 20 but less than more than 30 but less than more than 40 but less than more than 50 but less than Total

8 Note: In this example to classify the data we used intervals of 10. However, there is no rule for that. The decision should be case speci c Note: The best graph is always the one that displays the information in the most clear and apprehensible way. There is no restriction for the type of the graph that you would use. However, remember that it may also be risky not to use standard graphs as it may lead confusion for the readers Histogram It is a graph of the (numerical) data in a frequency distribution Interval Frequency 10 but less than but less than but less than but less than but less than 60 2 Frequency Histogram: Daily High Temperature

9 The Cumulative Frequency Distribution & Ogive (graphing cumulative frequencies) Interval Frequency Percentage Cumulative Cumulative Frequency Percentage more than 10 but less than more than 20 but less than more than 30 but less than more than 40 but less than more than 50 but less than Total Ogive: Daily High Temperature 100 Cumulative Percentage

10 A line chart (time-series plot) It is used to show the values of a variable over time (time series data) Time is measured on the horizontal axis The variable of interest is measured on the vertical axis An Example: Magazine Subscriptions by Year Thousands of subscribers Cross-Sectional Data: Refers to data collected by observing many subjects at the same point of time. It is collected usually for the purpose of comparison Time series-cross-sectional Data: Refers to data collected by observing many subjects at the successive points in time 10

11 The shape of the distribution The shape of the distribution is said to be symmetric if the observations are balanced, or evenly distributed, about the center Symmetric Distribution Frequency The shape of the distribution is said to be skewed if the observations are not symmetrically distributed around the center Negatively Skewed Distribution Positively Skewed Distribution Frequency Frequency

12 Tables and Graphs to Describe Relationship Between Variables Graphs illustrated so far have involved only a single variable When two variables exist other techniques are used: Categorical (Qualitative) Variables: Cross tables (or contingency tables) Numerical (Quantitative) Variables : Scatter plots Cross Tables If there are r categories for the rst variable (rows) and c categories for the second variable (columns), the table is called an r x c cross table Ex: 4 x 3 Cross Table for Investment Choices by Investor Investment Investor A Investor B Investor C Total Category Stocks Bonds CD Savings Total

13 Comparing Investors Savings CD Side by side bar chart Bonds Stocks Investor A Investor B Investor C Scatter Diagrams They are used for paired observations taken from two numerical variables. One variable is measured on the vertical axis and the other variable is measured on the horizontal axis Volume per day Cost per day Cost per Day Cost per Day vs. Production Volume Volume per Day 13

14 Chapter Summary Data (veri) in raw form are usually not easy to use for decision making. Some type of organization in the form of table or graphs is needed Terms reviewed in this chapter: Variables (De¼gişkenler): Categorical (Kategorik) Qualitative (Niteliksel) Independent (Ba¼g ms z) Numerical (Say sal) Quantitative (Niceliksel) Dependent (Ba¼g ml ) Ordinal scale (S rasal Ölçek) Ratio scale(oransal Ölçek) Interval scale(aral ksal Ölçek) Nominal scale (Say sal Ölçek) Line chart (Çizgisel gra k) Bar chart (Çubuk gra k) Pie chart (Dairesel Gra k) Pareto diagram (Pareto Diyagram ) Histogram (Histogram) Ogive (A cumulative line graph) The Cumulative Frequency distribution (Kümülatif Frekans Da¼g l m ) Time Series (Zaman Serisi) Time Series (Zaman Serisi) Skewed (Çarp k Da¼g l m) Scatter plot (Saç l m Gra ¼gi) 14

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16 CHAPTER 3: USING NUMERICAL MEASURES TO DESCRIBE DATA Measures of Central Tendency Mean: Arithmetic average of values (sum of values divided by the number of them) Median: Midpoint of ranked values Mode: Most frequently observed value in the data Ex: Suppose the following bicycle prices: 2.000, 100, 300, 100, 500 The mean is: ( )/5=600 The median can be found after ranking: 2.000, 500, 300, 100, 100; which is 300 The mode is 100 Even though the mean is the most generally used measure of central tendency, it is seen that it is subject to outliers that is, it is highly a ected from high or low values in the data even though these values may not be very informative Then median is often used, since the median is not sensitive to extreme values 16

17 Note: the location of the median is n position in the ordered data If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers Formally, the mean (also called arithmetic mean) is If calculated from population of N values, the mean is denoted by and calculated as: = NP x i i=1 N = x 1 + x 1 + ::: + x N N If calculated from sample size of n values, the mean is denoted by x and calculated as: x = np x i i=1 n = x 1 + x 1 + ::: + x n n 17

18 Mean and Median Depending on Shape of a Distribution Left Skewed Mean < Median Symmetric Mean = Median Right Skewed Median < Mean Measures of Variability Measures of variation give information on the spread or variability of the data values Ex: Same center, di erent variation 18

19 There are di erent measures of variability. The ones we are going to discuss Range: Di erence between the largest and the smallest observations Interquartile Range: Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data Variance: Average of squared deviations of values from the mean Standard Deviation: Square Root of Variance Coe cient of Variation: Standard Deviation divided by mean (shows relative variation) Range: Di erence between the largest and the smallest observations Ex: Range = 14 1 = 13 However, it ignores the way in which data are distributed and sensitive to outliers Range = 12 7 = Range = 12 7 = 5 19

20 Interquartile Range: Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data The rst quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger Q2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Ex: Median X X minimum Q1 (Q2) Q3 maximum 25% 25% 25% 25% Interquartile range = = 27 20

21 Variance: Average of squared deviations of values from the mean Population mean and variance = NP x i i=1 N 2 = NP (x i ) 2 i=1 N Sample mean and variance x = NP x i i=1 n s 2 = NP (x i x) 2 i=1 n 1 Standard Deviation: It is square root of variance. is the population standard deviation, and s is the sample standard deviation 21

22 Ex: Sample Data (x i ): 10, 12, 14, 15, 17, 18, 18, 24 The sample size, n = 8. The mean can be found by x = = 16 The standard deviation can be found by s s (10 x) 2 + (12 x) 2 + ::: + (24 x) 2 (10 16) 2 + (12 16) 2 + ::: + (24 16) 2 s = = n r 126 s = = 4:2426 (a measure of the average scatter around the mean) 7 You don t have to rank the data to nd variance or standard deviation Both measure is used for hypothesis testing for a single distribution, but cannot be used to compare variability of di erent distributions 22

23 Coe cient of Variation: Shows variation relative to mean, so that it measures relative variation and can be used to compare two or more sets of data measured in di erent units Ex: CV = ( s x )100% Stock A: Average price last year = $50 Standard deviation = $5 CV = ( 5 )100% = 10% 50 Stock B: Average price last year = $100 Standard deviation = $5 CV = ( 5 )100% = 5% 100 Both stocks have the same standard deviation, but stock B is less variable relative to its price 23

24 More About Standard Deviation of a Distribution Chebyshev s Theorem: For any distribution (not necessarily normal) with mean and standard deviation, and k > 1, the part of the observations that fall within the interval k (i.e. k standard deviations of the mean) includes at least this much of the data 100[1 (1=k 2 )]% Ex: At least Within (1 1=1:5 2 ) = 55:5 % k = 1:5 ( 1:5) (1 1=2 2 ) = 75 % k = 2 ( 2) (1 1=3 2 ) = 88:9 % k = 3 ( 3) 24

25 If the data distribution is bell-shaped (normally distributed), then the interval 1 contains about 68 % of the values in the population or the sample 2 contains about 95 % of the values in the population or the sample 3 contains about 99:7 % of the values in the population or the sample 68% μ μ ±1σ 95% μ ± 2σ 99.7% μ ± 3σ 25

26 Weighted Mean and Measures of Grouped Data np w i x i i=1 The weighted mean of a set of data is x = P wi = w 1x 1 + w 2 x 2 + ::: + w n x P n wi where w i is the weight of the i th observation Can be used when data is already grouped into n classes, with w i values in the i th class Suppose a data set contains values m 1 ; m 2 ; :::; m k, occurring with frequencies f 1 ; f 2 ; :::f K Population mean and variance = KP f i m i i=1 N where P N = K f i, and 2 = i=1 KP f i (m i ) 2 i=1 N Sample mean and variance x = NP f i m i i=1 n where P n = K f i, and s 2 = i=1 KP f i (m i x) 2 i=1 n 1 26

27 Measures of Relationships Between Variables The covariance measures the strength of the linear relationship between two variables Population covariance: Cov(x; y) = 2 xy = NP (x i x )(y i y ) i=1 N Sample covariance: Cov(x; y) = s 2 xy = np (x i x)(y i y) i=1 n 1 Only concerned with the strength of the relationship No causal e ect is implied Interpreting Covariance: Cov(x; y) > 0 ) x and y tend to move in the same direction Cov(x; y) < 0 ) x and y tend to move in opposite directions Cov(x; y) = 0 ) there is no linear relation between x and y 27

28 Coe cient of Correlation measures the relative strength of the linear relationship between two variables. It is relative because, unlike covariance, this measure is not a ected from the magnitude of data Population correlation coe cient: = Cov(x; y) x y Sample correlation coe cient: r = Cov(x; y) s x s y It is unit free and ranges between 1 and 1. The closer to 1, the stronger the negative linear relationship. 0 indicates no relationship between the variables of interest Y X X X r = 1 r =.6 r = 0 Y Y r = +1 X r = +.3 X r = 0 X 28

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30 Chapter Summary Terms reviewed in this chapter: Mean (Ortalama) Mode (Mod, Tepe De¼geri) Range (De¼gişim Aral ¼g ) Interquartile Range (Yar -çeyreklik De¼gişim Aral ¼g ) Standard Deviation (Standart Sapma) Covariance (Covaryasyon) Median (Medyan, Ortanca De¼ger) Measure (Ölçü) Variance (Varyasyon) Coe cient of Variation (Varyasyon Katsay s ) Weighted Mean (A¼g rl kl ortalama) Correlation (Corelasyon) 30