What is Statistics? Statistics is the science of understanding data and of making decisions in the face of variability and uncertainty.

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1 What is Statistics? Statistics is the science of understanding data and of making decisions in the face of variability and uncertainty. Statistics is a field of study concerned with the data collection, 1) organization, summarization and providing an overview of the general features of data, 2) and the drawing of inferences about a body of data (population) based on the properties of a part of the data (sample) observed. Population: the collection of all individuals of interest in a statistical study. Sample: any subset of individuals from the population. Branches of statistics Data 1) Descriptive statistics 2) Inferential statistics Data: the facts and figures that are collected, and analyzed in a statistical study. Individuals: are objects described by a set of data. [the entities on which data are collected] Variable: a characteristic of interest for the individual which takes on different values in different individual. Quantitative Variables (can be measured) [height, number of subscriptions,...] Qualitative (Categorical) Variables [hair color, gender,...] Discrete: a variable that has a countable number of possible values.(counts) Continuous: a variable that has an uncountable number of possible values.(measurements) Measurement Scales Nominal: consists of labels, names or categories. Ordinal: data that the order or rank is meaningful. Interval: numerical data that arithmetic operations are meaningful. Ratio: data that the ratio of two data is meaningful. Univariate data set consists of observations on a single variable. Multivariate data set consists of observations on more than one variable. Bivariate data set is a special multivariate case that consists of two variables.

2 Producing Data Type of Studies Observational Study: conditions to which subjects are exposed are not controlled by the investigator. (no attempt is made to control or influence the variables of interest) Experimental Study: conditions to which subjects are exposed to are controlled by the investigator. (treatments are used in order to observe the response) Completely Randomized Design: a design of experiment in which all the experimental units are allocated at random among all the treatments. Data Collection Techniques: Personal interview, Telephone interview, Self-administrated questionnaire (by person, mail, internet,...), Direct observation,... Sources of Data: Routinely kept records Surveys Designed Experiments External sources (internet, public or private organizations)... Preparing Data: Retain the raw data source Study log: data received, investigation, statisticians, data source Machine readable data base (save original files) Data cleaning (logic check, correcting, clarifying) Sampling Techniques: (To obtain representative sample) Simple random sampling:a sample obtained from a population in such a manner that all samples of the same size have equal likelihood of being selected. Stratified random sampling: Classify the population into at least two strata, then draw a sample from each. Cluster sampling: Divide the population into sections, randomly select few of those sections, and then choose all members in them. Systematic sample: Select every i-th member in the population according to their ID. Convenient sample: Use results that are already available or using data that is convenient to obtain.... Bias Response bias (influenced response, lies, ) Nonresponse bias

3 Descriptive Statistics Grouping, Describing and Displaying Data. Numerical Summary. [What type of statistical analysis is appropriate?] Categorical variable? Quantitative variable? Data Sheet ID Height Weight BirthMonth Exp. Gender H F H F T M H F T F H M H F H M T M H M T M T F H M T F H M H F T F T M H M T M H M H M Grouping and Displaying Categorical Data Frequency Table and Charts Class Frequency Relative Frequency Female 9 9/22 =.409 = 40.9% Male 13 13/22 =.591 = 59.1% Total % Female 40.9% Male 59.1% Pe rcent 10 0 Female Male sex

4 Grouping and Displaying Quantitative Data Frequency Table and Histogram Frequency Distribution Table (From data sheet) Class Frequency Relative Cumulative R.F. Freq < /22 =.091 2/ < /22 =.091 4/ < /22 =.136 7/ < /22 =.091 9/ < /22 = / < /22 = / < /22 = / < /22 = / < /22 = / < /22 = /22 Total Classes: Categories for grouping data. Frequency(class frequency): The number of data values in a class. Relative frequency: The ratio of the frequency of a class to the total number of pieces of data. Frequency distribution: A listing of classes and their frequencies. Relative Frequency distribution: A listing of classes and their relative frequencies. Upper class limit: The largest value that can go in a class. Lower class limit: The smallest value that can go in a class. Class width: The difference between the lower class limit of the given class and the lower class limit of the next higher class. Class midpoint (class mark): The midpoint of a class. Guidelines for grouping (quantitative) data: There should be between five and twenty classes. Each piece of data must belong to one, and only one, class. Whenever feasible, all classes should have the same width. Histogram (SPSS) Std. D ev = Mean = N = WEIGHT

5 (Excel ouput) What to observe in Histograms? Outliers: observations that stand out from the rest for some reason. Center: the middle of the data. Spread: the range; the extent of the data; how far the values are from each other. Shape: distribution pattern. [Skewness, symmetry, uniform,...] Symmetric (Bell) shape Skewed to the right, or positively skewed Bimodal Uniform Skewed to the left, or negatively skewed

6 Stemplots (or Stem-and-leaf plots) -- leading digits are called stems -- final digits are called leaves Splitting stemplots Back-to-back stemplots (What if the data are fractions?) Example: (number of hysterectomies performed by 15 male doctors) 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, For ten female doctors, the data are 5, 7, 10, 14, 18, 19, 25, 29, 31, Example: Number of hysterectomies performed by 15 male doctors: 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 for ten female doctors, the data are: 5, 7, 10, 14, 18, 19, 25, 29, 31, 33 (Male) (Female) Back-to-back stem-plot

7 Example: (Height data with gender) Female: 60, 63, 64, 65, 65, 65, 66, 67 Male: 61, 64, 69, 71, 71, 71, 72, 72, 72, 72, 73, 74, 75 (See data sheet) Female Male Back-to-back Female Male Split-back-to-back 430 6* 14 * => # 9 # => 5-9 7* # 5 Summarizing categorical data: Frequency distribution, relative frequency distribution,... Bar chart, Pie chart, Pareto chart,... Summarizing quantitative data: Frequency distribution, relative frequency distribution,... Histogram, Stemplot (Stem-and-leaf plot), box plot, dot plot,... When examining the data through graphs we should be on the alert for unusual values that do not follow the pattern of the rest of the data, some sense of a central or typical value of the data, some sense of how spread out or variable the data are, some sense of the shape of the overall pattern, in timeplots, be on the lookout for trends over time. Describing distribution with mathematical functions Density function, p(x) (Discrete distribution) x (Continuous distribution) Density, f (x) x

8 Examine Bivariate Data Categorical variables BW * SMOKE Crosstabulation SMOKE BW Normal Count Non-Smoke Smoke Total % of Total 79.0% 16.1% 95.2% Low birth weight Count % of Total 3.6% 1.2% 4.8% Total Count % of Total 82.6% 17.4% 100.0% Percent SMOKE 20 Non-Smoke 0 Normal Low birth weight Smoke Birth Weight Quantitative variables HEIGHT sex 62 Male 60 Female Total Population WEIGHT

9 Describing Distributions with Numbers Measuring center Mean: the average value of the data. If n observations are denoted by x 1, x 2,..., x n, their (sample) mean is x = x + x x n = n 1 2 n 1 n i= 1 x i Example: Birth weights (in lb) of 5 babies born from a group of women under certain diet. 7, 6, 8, 7, 7 => mean = ( ) / 5 = 35/5 = 7 [near the center of the data set] Example: (number of hysterectomies performed by 15 male doctors) 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 => mean = For ten female doctors, the data are 5, 7, 10, 14, 18, 19, 25, 29, 31, 33 mean =? Median: of a data set is the data value exactly in the middle of its ordered list if the number of pieces of data is odd, the mean of the two middle data values in its ordered list if the number of pieces of data is even. [median is not influenced by outliers and is best for nonsymmetric distribution] Example: (number of hysterectomies performed by 15 doctors) 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 ordered list => 20, 25, 25, 27, 28, 31, 33, 34, 36, 37, 44, 50, 59, 85, 86 median = 34 Example: (Birth weights for 6 infants.) 5, 7, 6, 8, 5, 9 ordered list => 5, 5, 6, 7, 8, 9 median = (6+7) / 2 = 6.5

10 Mode: of a data set is the observation that occurs most frequently. Example: (number of hysterectomies performed by 15 doctors) 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 ordered list => 20, 25, 25, 27, 28, 31, 33, 34, 36, 37, 44, 50, 59, 85, 86 Mode = 25 What is the Modal class? Measuring spread (variability) Range = largest data value smallest data value Sample from group I(diet program I): 7, 6, 8, 7, 7 => mean = ( ) / 5 = 35/5 = 7 Sample from group II(diet program II): 3, 4, 8, 9, 11=> mean = ( ) / 5 = 35/5 = 7 range of I = 8-6 = 2 range of II = 11-3 = 8 Is there any difference between the two samples? Quartiles: The first quartile, Q 1, or 25 th percentile, is the median of the lower half of the list of ordered observations. The third quartile, Q 3, or 75 th percentile, is the median of the upper half of the list of ordered observations. [percentiles?] Interquartile range (IQR) = Q 3 Q 1 Example: (data sheet) [odd number of data] 60,61,63,64,64,65,65,65,66,67,69,71,71,71,72,72,72,72,73,74,75 IQR = = 7.5 Q 1 = 64.5 Median = 69 Q 3 = 72 [even number of data] 6, 60,61,63,64,64,65,65,65,66,67, 69,71,71,71,72,72,72,72,73,74,75 IQR =? Q 1 =? Median =? Q 3 =?

11 Variance and Standard Deviation If n observations are denoted by x 1, x 2,..., x n, their variance and standard deviation are Sample Variance: s 2 = n i= 1 ( x i x) n 1 2, Sample Standard Deviation: s = n i= 1 ( x i x) n 1 2 Example: Birth weights (in lb) of 5 babies born from a group of women under diet program II. Weight data, x 3, 4, 8, 9, 11 Deviation from mean Absolute Deviation from mean = = = = = = = = = = 4 16 Total Square of deviation Sample Variance = 46/4 = 11.5 lb, Sample Standard Deviation = 46 / 4 = 3.39 lb. What is the standard deviation of the weights of babies from the sample of mothers who received diet program I? Does the mother s diet program affect the birth weights of babies? Coefficient of Variation (C.V.): is the standard deviation expressed as a percentage of the mean. It is a unit-free measure of dispersion. It provides a measurement for comparing relative variability of data sets from different scales. s 100 x C.V. = % Example: As part of the Berkeley Guidance Study. The heights (in cm) and weights (in kg) of 13 girls were measured at age two. The average height was 86.6 cm with a s.d.= 2.9 and average weight is 12.6 kg with a s.d.= 1.4. C.V. (height) = (2.9/86.6)100% = 3.3% C.V. (weight) = (1.4/12.6)100% = 11.1% More variability is weight than in height. Which of the following data has relatively higher variability? Systolic blood pressures: 113, 124, 125, 133, 146, 151, 169 Body temperature: 97.2, 97.5, 97.6, 97.9, 98.1, 98.2, 98.5

12 The five-number summary 1. Minimum value 2. Q 1 3. Median 4. Q 3 5. Maximum value Example: (data sheet without outlier 6 ) 80 The five-number summary and boxplot Min = 60, Q 1 = 64.5, Median = 69, Q 3 = 72, Max = 75. Boxplot (Box-and-Whisker Plot) N = 21 HEIGHT Interquartile range (IQR) = Q 3 Q 1 Inner and outer fences The inner fences are located at a distance of 1.5 IQR below Q 1 and at a distance of 1.5 IQR above Q 3. The outer fences are located at a distance of 3 IQR below Q 1 and at a distance of 3 IQR above Q 3. Mild and Extreme outliers Data values falling between the inner and outer fences are considered mild outliers. Data values falling outside the outer fences are considered extreme outliers. When outliers exist, the whisker extended to the smallest and largest data values within the inner fence.

13 N = 1 22 HEIGHT Side-by-side boxplot HEIGHT 50 N = 8 Female 13 Male sex

14 About s (sample standard deviation) : s measures the spread around the mean. the larger s is, the more spread out the data are. if s = 0, then all the observations must be equal. s is strongly influenced by outliers. Remarks: If the distribution of the data is symmetric, then the mean and median will be the same. The five-number summary is best for nonsymmetric data. The median and quartiles are not influenced by outliers. The mean and standard deviation are most appropriate to use only if the data are symmetric because both of these measures are easily influenced by outliers. Properties of a symmetric and bell-shaped (Normal) distribution: the distribution is symmetric about it mean (µ), 68% of the area is between µ σ and µ + σ, 95% of the area is between µ 2σ and µ + 2σ, 99.7% of the area is between µ 3σ and µ + 3σ. If x is an observation from a distribution that has mean µ, and standard deviation σ, the standardized value of x is, z-score of x : z = x µ σ µ + 3σ has a z-score 3, since it is 3 s.d. from mean. If a distribution has a mean 10 and a s.d. 2 and the value 7 has a z-score 1.5 = (7 10)/2. Chebychev s inequality There is at least 1 (1/k 2 ) of the data in a data set lie within k standard deviation of their mean. Example: Heart rates for asthmatic patients in a state of respiratory arrest has a mean of 140 beats per minute and a standard deviation of 35.5 beats per minute. With Chebychev s inequality, we know there are at least 75% of the patients that had heart rates between two standard deviations of the mean, (i.e., 140-2x35.5 = 69 and 140+2x35.5 = 211) in a state of respiratory arrest. Because, 1 (1/2 2 ) = ¾ = 75%.