Statistical POPULATION : -Collection of data we wish to gather information about - Eg: All students of CFS IIUM

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1 CHAPTER 3: INTODUCTION TO STATISTICS The Statistical Process Statistical POPULATION : -Collection of data we wish to gather information about - Eg: All students of CFS IIUM Make Inferences : Determine what the statistics tell us about the Population Plan the Investigation: What? How? Who? Where? Collect the Sample SAMPLE: Data collected from Population -Eg: Students of Dept. of Science Sample Statistics : -Graphic : Eg: Histogram, Ogive, Frequency Polygon -Numeric : Eg: Mean, Standard Deviation Analyze the Data : Organize, Describe & Present them 3.1 Introduction Statistics: A field of study which implies collecting, presenting, analyzing and interpreting data as a basis for explanation, description and comparison. used to analyze the results of surveys and as a tool in scientific research to make decisions based on controlled experiments. Also useful for operations, research, quality control, estimation and prediction. Population: a collection, or set of individuals or objects or events whose properties are to be analyzed. Sample: a group of subjects selected from the population. Sample is a subset of a population.

2 Data: consist a set of recorded observations or values. Any quantity that can have a number of values is variable. Variables whose values are determined by chance are called random variables. Data set: a collection of data values. Each value in the data set is called a data value or a datum. Variable: a characteristics or attribute that can assume different values. A statistical exercise normally consists of 4 stages: i) Collection of data by counting or measuring. ii) Ordering and presentation of the data in a convenient form. iii) Analysis of the collected data. iv) Interpretation of the results and conclusions formulated Two branches of Statistics

3 STATISTICS DESCRIPTIVE STATISTICS Consists of the collection, organization, summarization and presentation of data. -Describes a situation. Data presented in the form of charts, graphs or tables. -Make use of graphical techniques and numerical descriptive measures such as average to summarize and present the data. -E.g.: National census conducted by Malaysian goverment every 5 years or 10 years. The results of this census give some information regarding average age, income and other characteristics of the Malaysian population INFERENTIAL STATISTICS Consists of generalizing from samples to populations, performing hypothesis tests, detemining relationships among variables and making prediction - Inferences are made from samples to populations -Use probability, that is the chance of an event occurring. -The area of inferential statistics called hypotesis testing is a decision-making process for evaluating claims about a population, based on information obtined from samples. - E.g.: A researcher may want to know if a new product of skin lotion containing aloe vera will reduce the skin problem on children. For this study, two group of young children would be selected. One group would be given the lotion containing aloe vera and the other would be given a normal lotion without containing aloe vera. As aresult is observed by experts to see the effectiveness of the new product Variables and Types of Data LEVEL OF MEASUREMENT

4 NOMINAL QUALITATIVE ORDINAL TYPES OF DATA (VARIABLES) RATIO CONTINUOUS QUANTITATIVE INTERVAL DISCRETE Statisticians gain information about a particular situation by collecting data for random variables. Types of Data (variables) 1) Qualitative variables Variables that can be placed into distinct categories, according to some characteristics or attribute. Nonnumeric categories E.g.: Gender, color, religion, workplace and etc ) Quantitative variables It is numerical in nature and can be ordered or ranked. A quantitative variable may be one of two kinds: Discrete variable a variable that can be counted or for which there is a fixed set of values. Example: the number of children in a family, the number of students in a class and etc Continuous variable a variable that can be measured on continuous scale, the result depending on the precision of the measuring instrument, or the accuracy of the observer. Continuous variable can assume all values between any two specific values. Example: temperatures, heights, weights, time taken and etc. Variables can be classified by how they are categorized, counted or measured. Data/ variables can be classified according to the LEVEL OF MEASUREMENT as follows:

5 1) Nominal Level Data: - classifies data (persons/objects) into two or more categories. Whatever the basis for classification, a person can only be in one category and members of a given category have a common set of characteristics. The lowest level of measurement. No ranking/order can be placed on the data E.g. : Gender (Male / Female), Type of school (Public / Private), Height (Tall/Short), etc ) Ordinal Level Data:- classifies data into categories that can be ranked; however precise differences between the ranks do not exist. This type of measuring scale puts the data/subjects in order from highest to lowest, from most to least. It does not indicate how much higher or how much better. Intervals between ranks are not equal. E.g.: Letter grades (A,B,C,D,E,F) ; Man s build (small, medium, or large)-large variation exists among the individuals in each class. 3) Interval Level Data:- has all characteristics of a nominal and ordinal scale but in addition it is based upon predetermined equal interval. It has no true zero point (ratio between number on the scale are not meaningful). E.g.: Achievement test; aptitude tests, IQ test. A one point difference between IQ test of 110 and an IQ of 111 gives a significant difference. The Fahrenheit scale is a clear example of the interval scale of measurement. Thus, 60 degree Fahrenheit or -10 degrees Fahrenheit represent interval data. Measurement of Sea Level is another example of an interval scale. With each of these scales there are direct, measurable quantities with equality of units. In addition, zero does not represent the absolute lowest value. Rather, it is point on the scale with numbers both above and below it (for example, -10degrees Fahrenheit). 4) Ratio Level Data:- possesses all the characteristics of interval scale and in addition it has a meaningful (true zero point). True ratios exist when the same variable is measured on two different members of the population. The highest, most precise level of measurement. E.g.: Weight, number of calls received; height Data collection and Sampling Techniques

6 Sampling is the process of selecting a number of individuals for a study in such a way that the individuals represent the larger group from which they were selected. The purpose of sampling is to use a sample to gain information about a population. In order to obtain samples that are unbiased, statisticians use 4 basic methods of sampling: i) Random Sampling: subjects are selected by random numbers. ii) Systematic Sampling: Subjects are selected by using every kth number after the first subject is randomly from 1 through k. iii) Stratified Sampling: Subjects are selected by dividing up the population into groups (strata) and subjects within groups are randomly selected. - E.g.: We divide the population into 5 group then we take the subjects from each group to become our sample. iv) Cluster Sampling: Subjects are selected by using an intact group that is representative of the population. - E.g.: We divide the population into 5 group then we take groups to become our sample. That means group of subject represent 5 groups of subjects. Exercise: A ) Classify each set of data as discrete or continuous. 1) The number of suitcases lost by an airline. ) The height of corn plants. 3) The number of ears of corn produced. 4) The number of green M&M's in a bag. 5) The time it takes for a car battery to die. 6) The production of tomatoes by weight. B) Identify the following as nominal level, ordinal level, interval level, or ratio level data.

7 1) Percentage scores on a Math exam. ) Letter grades on an English essay. 3) Flavors of yogurt. 4) Instructors classified as: Easy, Difficult or Impossible. 5) Employee evaluations classified as : Excellent, Average, Poor. 6) Religions. 7) Political parties. 8) Commuting times to school. 9) Years (AD) of important historical events. 10) Ages (in years) of statistics students. 11) Ice cream flavor preference. 1) Amount of money in savings accounts. 13) Students classified by their reading ability: Above average, Below average, Normal. 3. ORGANIZING DATA & PRESENTATION OF DATA

8 3..1 FREQUENCY DISTRIBUTION A frequency distribution is the organization of raw data in table form, using classes and frequencies. There are three types of frequency distribution. 1. Categorical frequency distribution -for data that can be placed in specific category Example : The following data represent the color of men s shirts purchased in the men s department of a large department store. Construct a frequency distribution for the data. (W = White, BL = Blue, BR = Brown, Y =Yellow, G = Gray) W W BR Y BL BL W W Y G W W BL BR BL BR BL BL BR Y BL G W BL W W BL W BL BR Y BL G BR G BR W W BR Y W BL Y W W BL W BR G G (A complete categorical distribution must have class, frequency & percentage column in the table). Grouped frequency distribution -when the range of the data is large, the data must be grouped into classes. Example: The ages of the signers of the Declaration of Independence are shown below. Construct a frequency distribution for the data using seven classes Example: The number of calories per serving for selected ready-to-eat cereals is listed here. Construct a frequency distribution using seven classes Ungrouped frequency distribution -when the range of data is small

9 Example: A survey taken in a restaurant shows the following number of cups of coffee consumed with each meal. Construct frequency distribution Procedure to construct frequency distribution (this procedure is not unique): 1) Determine number of classes which normally 5 0 ) Find range = Highest value lowest value range 3) The class width should be an odd number. Class width = and rounding no. of class up. 4) Class Limit : Lower class limit =the lowest value or any number less than the lowest value. Upper class limit = (Lower class limit + class width) -1 5) Class Boundary: (to separate classes so that there are no gap in the frequency distribution) Lower Class Boundary: Lower class limit -0.5 Upper Class Boundary: Upper class limit ) Find frequency and cumulative frequency. Class width = Upper Class Boundary - Lower Class Boundary = Lower class limit of one class - Lower class limit of next class = Upper class limit of one class - Upper class limit of next class Class Midpoint = (Lower Class Boundary + Upper Class Boundary)/ = (Upper class limit + Lower class limit)/ 3.. HISTOGRAMS, FREQUENCY POLYGONS AND OGIVES

10 Example: For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct: Class Limit Frequency Histogram i) x-axis :class boundary ii) x-axis :class boundary y-axis : frequency y-axis : relative frequency. Frequency Polygon i) x-axis :class midpoint ii) x-axis :class midpoint y-axis : frequency y-axis : relative frequency 3. Ogive i) x-axis : class boundary ii) x-axis : class boundary y-axis : cumulative frequency y-axis : cumulative relative frequency f Relative frequency = f cumulative frequency Cumulative relative frequency = f each class to the total relative frequency. or add the relative frequency in Note: Graphing Given the frequency distribution below:

11 Class Limit Class Boundary f Cf The first value on the x-axis is -0.5 can be drawn as below OR All graphs must be drawn on the right side of y-axis and omit question on analyzing the graph in exercise. Exercise: 1. In a class of 35 students, the following grade distribution was found. Construct a histogram, frequency polygon and ogive for the data. (A=4, B=3, C=, D=1, F=0) Grade Frequency Using the histogram shown below. Construct i) A frequency distribution ii) A frequency polygon iii) An ogive y Class Boundaries x

12 3. The number of calories per serving for selected ready-to-eat cereals is listed here. Construct a histogram, frequency polygon and ogive for the data using relative frequency Below is a data set for the duration (in minutes) of a random sample of 4 longdistance phone calls: a) Construct a frequency distribution table for the data using the classes 1 to 5 6 to 10 etc. b) Construct a cumulative frequency distribution table and use it to draw up an ogive. 5. The following table refers to the 003 average income (in thousand Ringgit) per year for 0 employees of company A. Income ( 000 Frequency Ringgit) a) Draw the histogram and frequency polygon for the above data. b) Construct the cumulative frequency table. Hence, draw up an ogive for the above data.

13 3.3 DATA DESCRIPTION MEASURES OF CENTRAL TENDENCY Mean, median and Mode for Ungrouped data Mean (arithmetic average) Symbol for Sample: X Symbol for Population: μ (Syllabus focus on sample formula), Mean, X X n Median : (the middle point in ordered data set) - arrange the data in order, ascending or descending n 1 - select the middle point or use formula T, n is number of data. - Then, the median is: the value at location T (for odd number of data) the average of the value at location T and the value at location (T +1) (for even number of data) Mode : the value that occur most often in the data set Example: 1) The following data are the number of burglaries reported for a specific year for nine western Pennsylvania universities. Find mean, median and mode. 61, 11, 1, 3,, 30, 18, 3, 7 ) Twelve major earthquakes had Richter magnitudes shown here. Find mean, median and mode. 7.0, 6., 7.7, 8.0, 6.4, 6., 7., 5.4, 6.4, 6.5, 7., 5.4 3) The number of hospitals for the five largest hospital systems is shown here. Find mean, median and mode. 340, 75, 13, 59, 151

14 Mean, median and Mode for Ungrouped frequency distribution Mean, X f X f Median : - find cumulative frequency - Location of median f Mode : the value with the largest frequency Example: 4) A survey taken in a restaurant. This ungrouped frequency distribution of the number of cups of coffee consumed with each meal was obtained. Find mean, median and mode. Number of cups Frequency Mean, median and Mode for Grouped frequency distribution Mean, X f X m f where; X m =class midpoint (Student must show the working ie. Find midpoint and f X ) Median : - find cumulative frequency - find location of median class f f F - Median L c f Where; L=lower boundary of the median class F = cumulative frequency until the point L (before median class) f = frequency of the median class c =class width of median class m

15 Mode : - find location of modal class : class with the largest frequency - Mode c where; L=lower boundary of the modal class a = different between frequencies of modal class and the class before it. b= different between frequencies of modal class and the class after it. c =class width of median class Example: 5) These numbers of books were read by each of the 8 students in a literature class. Find mean, median and mode. Number of books Frequency ) Eighty randomly selected light bulbs were tested to determine their lifetimes (in hours). This frequency distribution was obtained. Find mean, median and mode. Class Boundaries Frequency

16 3.3. MEASURES OF VARIATION Variance and Standard deviation (the spread of data set) Group A Group B X =81 X =81 Variation, s =1 Variation, s = Even though the average for both groups is the same, the spread or variation of data in the Group B larger than Group A. Variance Population variance, σ = (Σ(X -μ) )/N Sample variance, s Standard deviation Population standard deviation, σ = (Σ(X -μ) )/N = σ Sample standard deviation, s (Syllabus focus on sample formula)

17 Sample variance and standard deviation For Ungrouped Data X X Variance, s Standard deviation, n 1 where; X =individual value X =sample mean n = sample size s s X X n 1 OR Variance, s (Note: X X n n 1 X is not the same as X Standard deviation, ) s s X X n n 1 Example: 1) The normal daily temperatures (in degrees Fahrenheit) in January for 10 selected cities are as follows. Find the variance and standard deviation ) Twelve students were given an arithmetic test and the times (in minutes) to complete it were Find the variance and standard deviation.

18 For Grouped Data Variance, s f X m Standard deviation, s f X m f f 1 s f X m f X m f f 1 (Students must show the working ie. Find f X and f X ) Example: 3) In a class of 9 students, this distribution of quiz scores was recorded. Find variance and standard deviation. m m Grade Frequency ) Eighty randomly selected light bulbs were tested to determine their lifetimes (in hours). This frequency distribution was obtained. Find variance and standard deviation. Class Boundaries Frequency ) These data represent the scores (in words per minute) of 5 typists on a speed test. Find variance and standard deviation. Class limit Frequency

19 3.3.3 MEASURES OF POSITION Standard scores, percentiles, deciles and quartiles are used to locate the relative position of the data value in the data set. Standard score / z-score The z-score represent the number of standard deviations the data value is above or below the mean. Example: z X s X if the z score is positive, the score is above the mean if the z score is negative, the score is below the mean 1) Let data set : 65, 70, 75,80, 85 ; X =75, s = X -s X - s X X +s X +s z= - z= -1 z= 0 z= 1 z= For data value 83: z ) Test marks are shown here. On which test she perform better? Math marks: ; X =53.3, s=10.4 Biology marks: ; X =75, s=5 z M z B z M z B, the relative position in math class is higher than her the relative position in biology class. She performs better in math paper than biology paper. (the marks that she get from biology paper is more than mathematics paper but we cannot compare the marks directly because the papers are different i.e. number of question, standard of questions and so on, that is why we have to compare the relative position)

20 Quartiles, deciles and percentile For Ungrouped data Quartiles: divide the distribution into four group Q1, Q, Q3 Smallest data Q1 Q Q3 Largest data 5% 5% 5% 5% Median arrange the data in order Find location of quartiles, nq c 4 where ; n = total number of values q =quartile i) If c is not whole number, round up to the next whole number ii) If c is a whole number, take average of c th and (c+1) th Example: 1) The weights in pounds in the data set. Find Q1, Q, Q ) The test score in the data set. Find Q1, Q, Q Deciles: divide the distribution into 10 groups Smallest data D1 D D3 D4 D5 D6 D7 D8 D9 Largest data 10% 10% 10% 10% 10% 10% 10% 10% 10% Median arrange the data in order Find location of quartiles, nd c 10 where ; n = total number of values d =decile iii) If c is not whole number, round up to the next whole number iv) If c is a whole number, take average of c th and (c+1) th

21 Example: 1) (from previous example) Find D ) (from previous example)find D Percentiles: divide the distribution into 100 equal groups Smallest data P1 P P3 P97 P98 P99 Largest data 10% 10% 10% 10% 10% 10% 10% 10% 10% D1, D, D3,, D9 correspond to P10, P0, P30,, P90 Q1, Q, Q3 correspond to P5, P50, P75 Median = Q = D5 = P50 arrange the data in order Find location of quartiles, np c 100 where ; n = total number of values p =percentile v) If c is not whole number, round up to the next whole number vi) If c is a whole number, take average of c th and (c+1) th Example: 1) (from previous example) Find P ) (from previous example)find P Finding percentile corresponding to given value, X number of values below X 0.5 Percentile 100% total number of values Example of data set : Find percentile for Percentile 100% 70% 5 P70 = 4 (round off the answer)

22 Example: ) (from previous example)find the percentile rank for each test score in the data set (Data value 47 = P64 but previously when we want to find P60 the data value is 47b too. So actually P60 closer to P64 which is data value 47) For Grouped Data METHOD 1: (USE PERCENTILE GRAPH) x-axis: class boundaries y-axis: relative cumulative frequency (percentage) Cumulative relative frequency (%) = cumulative frequency 100% f Graph: i) percentile graph Relative cumulative frequency (%) P5 ii) Ogive using relative frequency Relative cumulative frequency (iii) Ogive Cumulative Frequency P P5 5% x 75 =18.75

23 METHOD : (USE FORMULA) n f F P 100 n Ln c f Example: This distribution represents the data for weights of fifth-grade boys. Weights (pounds) frequency ) Find the approximate weights corresponding to each percentile given by constructing a percentile graph. (i) Q1 (ii) D8 (iii) Median (iv) P95 ) Find the approximate percentile ranks of the following weights. (i) 57 pounds (ii) 64 pounds (iii) 6 pounds (iv) 59 pounds 3) Find P63 by using the formula.

24 EXERCISE CHAPTER 3 1. What type of sampling is being employed if a country is divided into economic classes and a sample is chosen from each class to be surveyed?. Given a set of data 5,,8,14,10,5,7,10,m, n where X =7 and mode = 5. Find the possible values of m and n. (ans: m=5, n=4 or m =4, n =5) 3. Find the value that corresponds to the 30 th percentile of the following data set: (ans: P30 =8) 4. Given the variance of the set of 8 data x1, x, x3,, x8 is If X , find the mean of the data. (ans: 11.09) 5. Find Q3 for the given data set : 18,,50,15,13,6,5,1 (ans: 0) 6. The number of credits in business courses that eight applicants took is 9, 1, 15, 7, 33, p, 63, 7. Given the value that corresponds to the 75 th percentile is 54, find p. (ans: 45) 7. The mean of 5, 10, 6, 30, 45, 3, x, y is 5 where x and y are constants. If x = 16, find the median. (ans: 8) 8. A physician is interested in studying scheduling procedures. She questions 40 patients concerning the length of time in minutes that they waste past their scheduled appointment time. The following data are obtained: a) Construct a frequency distribution by using 7 classes (use 3 as lower limit of the first class) b) Find the mean, mode and standard deviation. (ans: 8.15, 31.3, 14.63) c) Draw an ogive by using relative frequency and estimate the median from the graph.