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- 1. QQS1013<br />ELEMENTARY STATISTIC<br />CHAPTER 2 <br />DESCRIPTIVE STATISTICS<br />2.1Introduction<br />2.2Organizing and Graphing Qualitative Data<br />2.3Organizing and Graphing Quantitative Data<br />2.4Central Tendency Measurement<br />2.5Dispersion Measurement<br />2.6Mean, Variance and Standard Deviation for Grouped Data<br />2.7Measure of Skewness<br />OBJECTIVES<br />After completing this chapter, students should be able to: <br />Create and interpret graphical displays involve qualitative and quantitative data.<br />Describe the difference between grouped and ungrouped frequency distribution, frequency and relative frequency, relative frequency and cumulative relative frequency.<br />Identify and describe the parts of a frequency distribution: class boundaries, class width, and class midpoint.<br />Identify the shapes of distributions.<br />Compute, describe, compare and interpret the three measures of central tendency: mean, median, and mode for ungrouped and grouped data.<br />Compute, describe, compare and interpret the two measures of dispersion: range, and standard deviation (variance) for ungrouped and grouped data.<br />Compute, describe, and interpret the two measures of position: quartiles and interquartile range for ungrouped and grouped data.<br />Compute, describe and interpret the measures of skewness: Pearson Coefficient of Skewness.<br />Introduction <br /> Raw data - Data recorded in the sequence in which there are collected and before they are processed or ranked. <br />Array data - Raw data that is arranged in ascending or descending order. <br />Example 1<br />Here is a list of question asked in a large statistics class and the “raw data” given by one of the students:<br />What is your sex (m=male, f=female)?<br />Answer (raw data): m<br />How many hours did you sleep last night?<br />Answer: 5 hours<br />Randomly pick a letter – S or Q.<br />Answer: S<br />What is your height in inches?<br />Answer: 67 inches<br />What’s the fastest you’ve ever driven a car (mph)?<br />Answer: 110 mph<br />Example 2<br />Quantitative raw data<br />Qualitative raw data <br />These data also called ungrouped data<br />2.2Organizing and Graphing Qualitative Data <br />2.2.1Frequency Distributions/ Table<br />2.2.2Relative Frequency and Percentage Distribution <br />2.2.3Graphical Presentation of Qualitative Data<br />2.2.1Frequency Distributions / Table<br />A frequency distribution for qualitative data lists all categories and the number of elements that belong to each of the categories.<br />It exhibits the frequencies are distributed over various categories<br />Also called as a frequency distribution table or simply a frequency table.<br />The number of students who belong to a certain category is called the frequency of that category.<br />457200185420<br />Relative Frequency and Percentage Distribution<br />A relative frequency distribution is a listing of all categories along with their relative frequencies (given as proportions or percentages).<br />It is commonplace to give the frequency and relative frequency distribution together.<br />Calculating relative frequency and percentage of a category<br />Relative Frequency of a category<br />= Frequency of that category<br /> Sum of all frequencies<br />Percentage = (Relative Frequency)* 100<br />Example 3<br />A sample of UUM staff-owned vehicles produced by Proton was identified and the make of each noted. The resulting sample follows (W = Wira, Is = Iswara, Wj = Waja, St = Satria, P = Perdana, Sv = Savvy):<br />WWPIsIsPIsWStWjIsWWWjIsWWIsWWjWjIsWjSvWWWWjStWWjSvWIsPSvWjWjWWStWWWWStStPWjSv<br />Construct a frequency distribution table for these data with their relative frequency and percentage.<br />Solution:<br />CategoryFrequencyRelative FrequencyPercentage (%)Wira1919/50 = 0.380.38*100= 38Iswara80.1616Perdana40.088Waja100.2020Satria50.1010Savvy40.088Total501.00100<br />Graphical Presentation of Qualitative Data<br />Bar Graphs<br />A graph made of bars whose heights represent the frequencies of respective categories.<br />Such a graph is most helpful when you have many categories to represent.<br />Notice that a gap is inserted between each of the bars.<br />It has<br />=>simple/ vertical bar chart<br />=> horizontal bar chart <br />=> component bar chart <br />=> multiple bar chart<br />Simple/ Vertical Bar Chart<br />To construct a vertical bar chart, mark the various categories on the horizontal axis and mark the frequencies on the vertical axis<br />Refer to Figure 2.1 and Figure 2.2,<br />27432004445<br />Figure 2.1 Figure 2.2<br />Horizontal Bar Chart<br />To construct a horizontal bar chart, mark the various categories on the vertical axis and mark the frequencies on the horizontal axis.<br />Example 4: Refer Example 3, <br />left15240<br /> <br />Figure 2.3<br />Another example of horizontal bar chart: Figure 2.4<br />center635<br />Figure 2.4: Number of students at Diversity College who are immigrants, by last country of permanent residence <br />Component Bar Chart <br />To construct a component bar chart, all categories is in one bar and every bar is divided into components. <br />The height of components should be tally with representative frequencies.<br />Example 5<br />Suppose we want to illustrate the information below, representing the number of people participating in the activities offered by an outdoor pursuits centre during Jun of three consecutive years.<br />200420052006Climbing213436Caving101221Walking7585100Sailing363640Total142167191<br />Solution:<br />Figure 2.5<br />Mulztiple Bar Chart<br />To construct a multiple bar chart, each bars that representative any categories are gathered in groups. <br />The height of the bar represented the frequencies of categories.<br />Useful for making comparisons (two or more values). <br />Example 6: Refer example 5,<br />center165100Figure 2.6<br />Another example of horizontal bar chart: Figure 2.7<br />457200100330<br />Figure 2.7: Preferred snack choices of students at UUM<br />The bar graphs for relative frequency and percentage distributions can be drawn simply by marking the relative frequencies or percentages, instead of the class frequencies. <br />Pie Chart<br />A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories.<br />An alternative to the bar chart and useful for summarizing a single categorical variable if there are not too many categories.<br />The chart makes it easy to compare relative sizes of each class/category.<br />The whole pie represents the total sample or population. The pie is divided into different portions that represent the different categories.<br />To construct a pie chart, we multiply 360o by the relative frequency for each category to obtain the degree measure or size of the angle for the corresponding categories.<br />Example 7 (Table 2.6 and Figure 2.8):<br />0482603314700581660<br />Table 2.6 Figure 2.8<br />Example 8 (Table 2.7 and Figure 2.9):<br />Movie GenresFrequencyRelative FrequencyAngle SizeComedyActionRomanceDramaHorrorForeignScience Fiction543628282216160.270.180.140.140.110.080.08 360*0.27=97.2o 360*0.18=64.8o360*0.14=50.4o360*0.14=50.4o360*0.11=39.6o360*0.08=28.8o360*0.08=28.8o2001.00360o<br />left24765<br />Figure 2.9Figure 2.9<br />Line Graph/Time Series Graph<br />A graph represents data that occur over a specific period time of time.<br />Line graphs are more popular than all other graphs combined because their visual characteristics reveal data trends clearly and these graphs are easy to create. <br />When analyzing the graph, look for a trend or pattern that occurs over the time period. <br />Example is the line ascending (indicating an increase over time) or descending (indicating a decrease over time).<br />Another thing to look for is the slope, or steepness, of the line. A line that is steep over a specific time period indicates a rapid increase or decrease over that period.<br />Two data sets can be compared on the same graph (called a compound time series graph) if two lines are used. <br />Data collected on the same element for the same variable at different points in time or for different periods of time are called time series data.<br />A line graph is a visual comparison of how two variables—shown on the x- and y-axes—are related or vary with each other. It shows related information by drawing a continuous line between all the points on a grid. <br />Line graphs compare two variables: one is plotted along the x-axis (horizontal) and the other along the y-axis (vertical). <br />The y-axis in a line graph usually indicates quantity (e.g., RM, numbers of sales litres) or percentage, while the horizontal x-axis often measures units of time. As a result, the line graph is often viewed as a time series graph<br />Example 9<br />A transit manager wishes to use the following data for a presentation showing how Port Authority Transit ridership has changed over the years. Draw a time series graph for the data and summarize the findings.<br />YearRidership(in millions)1990199119921993199488.085.075.776.675.4<br /> <br />Solution:<br />The graph shows a decline in ridership through 1992 and then leveling off for the years 1993 and 1994.<br />Exercise 1<br />The following data show the method of payment by 16 customers in a supermarket checkout line. Here, C = cash, CK = check, CC = credit card, D = debit and O = other.<br />CCKCKCCCDOCCKCCDCCCCKCKCC<br />Construct a frequency distribution table.<br />Calculate the relative frequencies and percentages for all categories.<br />Draw a pie chart for the percentage distribution.<br />The frequency distribution table represents the sale of certain product in ZeeZee Company. Each of the products was given the frequency of the sales in certain period. Find the relative frequency and the percentage of each product. Then, construct a pie chart using the obtained information.<br />Type of ProductFrequencyRelative FrequencyPercentageAngle SizeABCDE13125911<br />Draw a time series graph to represent the data for the number of worldwide airline fatalities for the given years.<br />Year1990199119921993199419951996No. of fatalities4405109908017325571132<br />A questionnaire about how people get news resulted in the following information from 25 respondents (N = newspaper, T = television, R = radio, M = magazine). <br />NNRTTRNTMRMMNRNTRMNMTRRNN<br />Construct a frequency distribution for the data.<br />Construct a bar graph for the data.<br />The given information shows the export and import trade in million RM for four months of sales in certain year. Using the provided information, present this data in component bar graph. <br />MonthExportImportSeptemberOctoberNovemberDecember2830322420281714<br />The following information represents the maximum rain fall in millimeter (mm) in each state in Malaysia. You are supposed to help a meteorologist in your place to make an analysis. Based on your knowledge, present this information using the most appropriate chart and give your comment.<br />StateQuantity (mm)PerlisKedahPulau PinangPerakSelangorWilayah Persekutuan Kuala LumpurNegeri SembilanMelakaJohorPahangTerengganuKelantanSarawakSabah435512163721664100339022387610501255986878456<br />2.3Organizing and Graphing Quantitative Data<br />2.3.1Stem and Leaf Display<br />2.3.2Frequency Distribution<br />2.3.3Relative Frequency and Percentage Distributions.<br />2.3.4 Graphing Grouped Data<br />2.3.5Shapes of Histogram<br />2.3.6Cumulative Frequency Distributions.<br />Stem-and-Leaf Display<br />In stem and leaf display of quantitative data, each value is divided into two portions – a stem and a leaf. Then the leaves for each stem are shown separately in a display.<br />Gives the information of data pattern.<br />Can detect which value frequently repeated.<br />Example 10<br />12 9 10 5 12 23 7<br /> 13 11 12 31 28 37 6<br />41 38 44 13 22 18 19<br />Solution:<br />09 5 7 6<br />12 0 2 3 1 2 4 3 8 9<br />25 3 8 2<br />36 1 7 8<br />41 4<br />Frequency Distributions<br />A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class.<br />Data presented in form of frequency distribution are called grouped data.<br />0163830<br />The class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class. Also called real class limit.<br />To find the midpoint of the upper limit of the first class and the lower limit of the second class, we divide the sum of these two limits by 2. <br />e.g.:<br />class boundary <br />Class Width (class size)<br /> Class width = Upper boundary – Lower boundary<br />e.g. :<br /> Width of the first class = 600.5 – 400.5 = 200<br />Class Midpoint or Mark<br /> <br />e.g:<br />Constructing Frequency Distribution Tables<br />1.To decide the number of classes, we used Sturge’s formula, which is<br />c = 1 + 3.3 log n<br /> where c is the no. of classes<br /> n is the no. of observations in the data set.<br />2. Class width,<br />This class width is rounded to a convenient number.<br />3.Lower Limit of the First Class or the Starting Point<br />Use the smallest value in the data set.<br />Example 11<br />The following data give the total home runs hit by all players of each of the 30 Major League Baseball teams during 2004 season<br />Solution:<br />Number of classes, c = 1 + 3.3 log 30<br /> = 1 + 3.3(1.48) <br />= 5.89 6 class<br />Class width, <br />Starting Point = 135<br />Table 2.10 Frequency Distribution for Data of Table 2.9<br />Total Home RunsTallyf135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242|||| |||||||||| |||| ||||||||1025634<br />Relative Frequency and Percentage Distributions<br />Example 12 (Refer example 11)<br />Table 2.11: Relative Frequency and Percentage Distributions <br />Total Home RunsClass BoundariesRelative Frequency%135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.50.33330.06670.16670.20.10.133333.336.6716.67201013.33Sum1.0100%<br />Graphing Grouped Data<br />Histograms<br />A histogram is a graph in which the class boundaries are marked on the horizontal axis and either the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies or percentages are represented by the heights of the bars.<br />In histogram, the bars are drawn adjacent to each other and there is a space between y axis and the first bar.<br />Example 13 (Refer example 11)<br />134.5 152.5 170.5 188.5 206.5 224.5 242.5<br />Figure 2.10: Frequency histogram for Table 2.10<br />Polygon<br />A graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines is called a polygon.<br />Example 13<br />34290082550<br />134.5 152.5 170.5 188.5 206.5 224.5 242.5<br />134.5 152.5 170.5 188.5 206.5 224.5 242.5Figure 2.11: Frequency polygon for Table 2.10<br />For a very large data set, as the number of classes is increased (and the width of classes is decreased), the frequency polygon eventually becomes a smooth curve called a frequency distribution curve or simply a frequency curve.<br />Figure 2.12: Frequency distribution curve<br /> Shape of Histogram <br />Same as polygon.<br />For a very large data set, as the number of classes is increased (and the width of classes is decreased), the frequency polygon eventually becomes a smooth curve called a frequency distribution curve or simply a frequency curve.<br />The most common of shapes are:<br />(i) Symmetric<br />Figure 2.13 & 2.14: Symmetric histograms<br />(ii) Right skewed and (iii) Left skewed<br />Figure 2.15 & 2.16: Right skewed and Left skewed<br />Describing data using graphs helps us insight into the main characteristics of the data.<br />When interpreting a graph, we should be very cautious. We should observe carefully whether the frequency axis has been truncated or whether any axis has been unnecessarily shortened or stretched.<br />Cumulative Frequency Distributions<br />A cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class.<br />Example 14: Using the frequency distribution of table 2.11,<br /> <br />Total Home RunsClass BoundariesCumulative Frequency135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.51010+2=1210+2+5=1710+2+5+6=2310+2+5+6+3=2610+2+5+6+3+4=30<br />Ogive<br />An ogive is a curve drawn for the cumulative frequency distribution by joining with straight lines the dots marked above the upper boundaries of classes at heights equal to the cumulative frequencies of respective classes.<br />Two type of ogive:<br />(i) ogive less than<br />(ii)ogive greater than<br />First, build a table of cumulative frequency.<br />Example 15 (Ogive Less Than)<br />56633730 – 3940 – 4950 – 5960 - 6970 – 7980 - 8930Number of students (f)TotalEarnings (RM)Earnings (RM) Cumulative Frequency (F)Less than 29.5Less than 39.5Less than 49.5Less than 59.5Less than 69.5Less than 79.5Less than 89.5051117202330<br />0510152025303529.539.549.559.569.579.589.5EarningsCumulative Frequency<br />Figure 2.17<br />Example 16 (Ogive Greater Than)<br />56633730 – 3940 – 4950 – 5960 - 6970 – 7980 - 8930Number of students (f)TotalEarnings (RM)<br /> Cumulative Frequency (F)Earnings (RM)<br />302519131070More than 29.5More than 39.5More than 49.5More than 59.5More than 69.5More than 79.5More than 89.5<br />0510152025303529.539.549.559.569.579.589.5EarningsCumulative Frequency<br /> Figure 2.18<br />Figure 2.18<br />2.3.7Box-Plot<br />Describe the analyze data graphically using 5 measurement: smallest value, first quartile (K1), second quartile (median or K2), third quartile (K3) and largest value.<br />Smallest valueLargest value K1 Median K3Largest value K1 Median K3Largest value K1 Median K3Smallest valueSmallest valueFor symmetry data For left skewed data For right skewed data<br />2.4Measures of Central Tendency <br />2.4.1 Ungrouped Data<br />(1) Mean<br />(2) Weighted mean<br />(3) Median<br />(4) Mode<br />Grouped Data<br />(1) Mean<br />(2) Median<br />(3) Mode<br /> Relationship among mean, median & mode<br />2.4.1 Ungrouped Data <br />Mean <br />Mean for population data: <br /> <br />Mean for sample data: <br />where: = the sum af all values <br /> N = the population size <br /> n = the sample size, <br /> µ = the population mean<br /> = the sample mean<br />Example 17<br />The following data give the prices (rounded to thousand RM) of five homes sold recently in Sekayang.<br />158189265127191<br />Find the mean sale price for these homes.<br />Solution:<br />Thus, these five homes were sold for an average price of RM186 thousand @ RM186 000.<br />The mean has the advantage that its calculation includes each value of the data set.<br />Weighted Mean<br />Used when have different needs.<br />Weight mean :<br /> <br />where w is a weight.<br />Example 18<br />Consider the data of electricity components purchasing from a factory in the table below:<br />TypeNumber of component (w)Cost/unit (x)12345120050025001000800RM3.00RM3.40RM2.80RM2.90RM3.25Total6000<br />Solution:<br />Mean cost of a unit of the component is RM2.97<br />Median<br />Median is the value of the middle term in a data set that has been ranked in increasing order.<br />Procedure for finding the Median<br />Step 1: Rank the data set in increasing order.<br />Step 2: Determine the depth (position or location) of the median.<br />Step 3: Determine the value of the Median. <br />Example 19<br />Find the median for the following data:<br />10 5 19 8 3<br />Solution:<br />(1) Rank the data in increasing order<br />3 5 8 10 19<br /> (2) Determine the depth of the Median <br /> <br /> <br /> (3) Determine the value of the median<br />Therefore the median is located in third position of the data set.<br />3 5 8 10 19<br /> Hence, the Median for above data = 8<br />Example 20<br />Find the median for the following data:<br />10 5 19 8 3 15<br />Solution:<br />(1) Rank the data in increasing order<br /> 35810 15 19<br /> <br />(2) Determine the depth of the Median <br /> <br />(3) Determine the value of the Median<br />Therefore the median is located in the middle of 3rd position and 4th position of the data set.<br />Hence, the Median for the above data = 9<br />The median gives the center of a histogram, with half of the data values to the left of (or, less than) the median and half to the right of (or, more than) the median.<br />The advantage of using the median is that it is not influenced by outliers.<br />Mode<br />Mode is the value that occurs with the highest frequency in a data set.<br />Example 21<br />1. What is the mode for given data?<br /> 77 69 74 81 71 68 74 73<br />2. What is the mode for given data?<br /> 77 69 68 74 81 71 68 74 73<br />Solution:<br />1. Mode = 74 (this number occurs twice): Unimodal <br />2.Mode = 68 and 74: Bimodal<br />A major shortcoming of the mode is that a data set may have none or may have more than one mode. <br />One advantage of the mode is that it can be calculated for both kinds of data, quantitative and qualitative.<br />Grouped Data <br />Mean<br />Mean for population data:<br />Mean for sample data:<br />Where the midpoint and f is the frequency of a class.<br />Example 22<br />The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean.<br />Numberof orderf10 – 1213 – 1516 – 1819 – 214122014 n = 50<br />Solution:<br />Because the data set includes only 50 days, it represents a sample. The value of is calculated in the following table:<br />Numberof orderfxfx10 – 1213 – 1516 – 1819 – 2141220141114172044168340280 n = 50 = 832<br />The value of mean sample is:<br />Thus, this mail-order company received an average of 16.64 orders per day during these 50 days.<br />Median<br />Step 1: Construct the cumulative frequency distribution.<br />Step 2: Decide the class that contain the median.<br />Class Median is the first class with the value of cumulative frequency is at least n/2.<br />Step 3: Find the median by using the following formula: <br />Where:<br />n = the total frequency<br />F = the total frequency before class median<br />i = the class width<br /> = the lower boundary of the class median<br /> = the frequency of the class median<br />Example 23<br />Based on the grouped data below, find the median:<br />Time to travel to workFrequency1 – 1011 – 2021 – 3031 – 4041 – 508141297<br />Solution:<br />1st Step: Construct the cumulative frequency distribution <br />Time to travel to workFrequency Cumulative Frequency1 – 1011 – 2021 – 3031 – 4041 – 508141297822344350<br /> Class median is the 3rd class<br />So, F = 22, = 12, = 21.5 and i = 10 <br />Therefore,<br />Thus, 25 persons take less than 24 minutes to travel to work and another 25 persons take more than 24 minutes to travel to work.<br />Mode<br />Mode is the value that has the highest frequency in a data set.<br />For grouped data, class mode (or, modal class) is the class with the highest frequency.<br />To find mode for grouped data, use the following formula:<br />Where:<br /> is the lower boundary of class mode<br />is the difference between the frequency of class mode and the frequency of the class before the class mode<br />is the difference between the frequency of class mode and the frequency of the class after the class mode<br /> i is the class width<br />Example 24<br />Based on the grouped data below, find the mode<br />Time to travel to workFrequency1 – 1011 – 2021 – 3031 – 4041 – 508141297<br />Solution: <br />Based on the table,<br /> = 10.5, = (14 – 8) = 6, = (14 – 12) = 2 and i = 10<br />We can also obtain the mode by using the histogram;<br />Figure 2.19<br />2.4.3 Relationship among mean, median & mode<br />As discussed in previous topic, histogram or a frequency distribution curve can assume either skewed shape or symmetrical shape.<br />Knowing the value of mean, median and mode can give us some idea about the shape of frequency curve.<br />For a symmetrical histogram and frequency curve with one peak, the value of the mean, median and mode are identical and they lie at the center of the distribution.(Figure 2.20) <br />2971800961390228600951865For a histogram and a frequency curve skewed to the right, the value of the mean is the largest that of the mode is the smallest and the value of the median lies between these two.<br />Figure 2.20: Mean, median, and mode for a symmetric histogram and frequency distribution curve <br />Figure 2.21: Mean, median, and mode for a histogram and frequency distribution curve skewed to <br />the right<br />2971800168275<br />For a histogram and a frequency curve skewed to the left, the value of the mean is the smallest and that of the mode is the largest and the value of the median lies between these two.<br />Figure 2.22: Mean, median, and mode for a histogram and frequency distribution curve skewed to the left<br />2.5Dispersion Measurement<br />The measures of central tendency such as mean, median and mode do not reveal the whole picture of the distribution of a data set.<br />Two data sets with the same mean may have a completely different spreads.<br />The variation among the values of observations for one data set may be much larger or smaller than for the other data set.<br />2.5.1 Ungrouped data<br />(1) Range<br />(2) Standard Deviation<br />2.5.2 Grouped data<br />(1) Range<br />(2) Standard deviation<br />2.5.3 Relative Dispersion Measurement<br />Ungrouped Data<br />Range<br />RANGE = Largest value – Smallest value<br />Example 25:<br />Find the range of production for this data set,<br />Solution: <br />Range = Largest value – Smallest value<br /> = 267 277 – 49 651<br /> = 217 626<br />Disadvantages:<br />being influenced by outliers.<br />Based on two values only. All other values in a data set are ignored.<br />Variance and Standard Deviation<br />Standard deviation is the most used measure of dispersion.<br />A Standard Deviation value tells how closely the values of a data set clustered around the mean.<br />Lower value of standard deviation indicates that the data set value are spread over relatively smaller range around the mean.<br />Larger value of data set indicates that the data set value are spread over relatively larger around the mean (far from mean).<br />Standard deviation is obtained the positive root of the variance: <br />VarianceStandard DeviationPopulationSample<br />Example 26<br />Let x denote the total production (in unit) of company<br />CompanyProductionABCDE62931267534<br />Find the variance and standard deviation,<br />Solution:<br />CompanyProduction (x)x2ABCDE629312675343844864915 87656251156 1156 <br /> <br />Since s2 = 1182.50;<br />Therefore,<br />The properties of variance and standard deviation:<br />(1) The standard deviation is a measure of variation of all values from the mean.<br />(2)The value of the variance and the standard deviation are never negative. Also, larger values of variance or standard deviation indicate greater amounts of variation.<br />(3)The value of s can increase dramatically with the inclusion of one or more outliers.<br />(4) The measurement units of variance are always the square of the measurement units of the original data while the units of standard deviation are the same as the units of the original data values.<br />Grouped Data <br />Range<br />Range = Upper bound of last class – Lower bound of first class<br />ClassFrequency41 – 5051 – 6061 – 7071 – 8081 – 9091 - 10013713106 Total40<br />Upper bound of last class = 100.5<br />Lower bound of first class = 40.5<br />Range = 100.5 – 40.5 = 60<br />Variance and Standard Deviation<br />VarianceStandard DeviationPopulationSample<br />Example 27<br />Find the variance and standard deviation for the following data:<br />No. of orderf10 – 1213 – 1516 – 1819 – 214122014 Totaln = 50<br />Solution:<br />No. of orderfxfxfx210 – 1213 – 1516 – 1819 – 2141220141114172044168340280484235257805600 Totaln = 5085714216<br />Variance, Standard Deviation, <br /> <br />320040048260 <br /> <br /> <br />Thus, the standard deviation of the number of orders received at the office of this mail-order company during the past 50 days is 2.75.<br />2.5.3 Relative Dispersion Measurement<br />To compare two or more distribution that has different unit based on their dispersion Or <br />To compare two or more distribution that has same unit but big different in their value of mean.<br />Also called modified coefficient or coefficient of variation, CV.<br /> <br /> <br />Example 28<br />Given mean and standard deviation of monthly salary for two groups of worker who are working in ABC company- Group 1: 700 & 20 and Group 2 :1070 & 20. Find the CV for every group and determine which group is more dispersed.<br />Solution:<br />The monthly salary for group 1 worker is more dispersed compared to group 2.<br />Measure of Position<br />Determines the position of a single value in relation to other values in a sample or a population data set.<br />Ungrouped Data<br />Quartiles<br />Interquatile Range<br />Grouped Data<br />Quartile<br />Interquartile Range<br />Quartiles<br />Quartiles are three summary measures that divide ranked data set into four equal parts.<br />The 1st quartiles – denoted as Q1<br />The 2nd quartiles – median of a data set or Q2<br />The 3rd quartiles – denoted as Q3<br />Example 29<br />Table below lists the total revenue for the 11 top tourism company in Malaysia<br />109.7 79.9 21.2 76.4 80.2 82.1 79.4 89.3 98.0 103.5 86.8<br />Solution:<br />Step 1: Arrange the data in increasing order<br />76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7 121.2<br />Step 2: Determine the depth for Q1 and Q3<br />Step 3: Determine the Q1 and Q3<br />76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7 121.2<br />Q1 = 79.9<br />Q3 = 103.5<br />Table below lists the total revenue for the 12 top tourism company in Malaysia<br />109.7 79.9 74.1 121.2 76.4 80.2 82.1 79.4 89.3 98.0 103.5 86.8<br />Solution:<br />Step 1: Arrange the data in increasing order<br />74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7 121.2<br />Step 2: Determine the depth for Q1 and Q3<br />Step 3: Determine the Q1 and Q3<br />74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7 121.2<br />Q1 = 79.4 + 0.25 (79.9 – 79.4) = 79.525<br />Q3 = 98.0 + 0.75 (103.5 – 98.0) = 102.125<br />Interquartile Range<br />The difference between the third quartile and the first quartile for a data set.<br /> IQR = Q3 – Q1 <br />Example 30<br />By referring to example 29, calculate the IQR.<br />Solution:<br />IQR = Q3 – Q1 = 102.125 – 79.525 = 22.6<br />2.6.2 Grouped Data<br />Quartiles<br />From Median, we can get Q1 and Q3 equation as follows:<br /> <br /> ; <br />Example 31<br />Refer to example 23, find Q1 and Q3<br />Solution:<br />1st Step: Construct the cumulative frequency distribution <br />Time to travel to workFrequency Cumulative Frequency1 – 1011 – 2021 – 3031 – 4041 – 508141297822344350<br />2nd Step: Determine the Q1 and Q3<br />Class Q1 is the 2nd class<br />Therefore,<br />Class Q3 is the 4th class<br />Therefore,<br />Interquartile Range<br />IQR = Q3 – Q1<br />Example 32:<br />Refer to example 31, calculate the IQR.<br />Solution:<br />IQR = Q3 – Q1 = 34.3889 – 13.7143 = 20.6746<br />Measure of Skewness<br />To determine the skewness of data (symmetry, left skewed, right skewed)<br />Also called Skewness Coefficient or Pearson Coefficient of Skewness <br />If Sk +ve right skewed <br />If Sk -ve left skewed<br />If Sk = 0 symmetry<br />If Sk takes a value in between (-0.9999, -0.0001) or (0.0001, 0.9999) approximately symmetry.<br />Example 33<br />The duration of cancer patient warded in Hospital Seberang Jaya recorded in a frequency distribution. From the record, the mean is 28 days, median is 25 days and mode is 23 days. Given the standard deviation is 4.2 days.<br />What is the type of distribution?<br />Find the skewness coefficient<br />Solution:<br />This distribution is right skewed because the mean is the largest value<br />So, from the Sk value this distribution is right skewed.<br />Exercise 2:<br />A survey research company asks 100 people how many times they have been to the dentist in the last five years. Their grouped responses appear below.<br />Number of VisitsNumber of Responses0 – 4165 – 92510 – 144815 – 1911<br />What are the mean and variance of the data?<br /> A researcher asked 25 consumers: “How much would you pay for a television adapter that provides Internet access?” Their grouped responses are as follows:<br />Amount ($)Number of Responses0 – 992100 – 1992200 – 2493250 – 2993300 – 3496350 – 3993400 – 4994500 – 9992<br />Calculate the mean, variance, and standard deviation.<br />The following data give the pairs of shoes sold per day by a particular shoe store in the last 20 days.<br />85 90 89 70 79 80 83 83 75 76<br />89 86 71 76 77 89 70 65 90 86<br />Calculate the<br />mean and interpret the value.<br />median and interpret the value.<br />mode and interpret the value.<br />standard deviation.<br />4. The followings data shows the information of serving time (in minutes) for 40 customers in a post office:<br />2.04.52.52.94.22.93.52.83.22.94.03.03.82.52.33.52.13.13.64.34.72.64.13.14.62.85.12.72.64.43.53.02.73.92.92.92.53.73.32.4<br />Construct a frequency distribution table with 0.5 of class width.<br />Construct a histogram.<br />Calculate the mode and median of the data.<br />Find the mean of serving time.<br />Determine the skewness of the data.<br />Find the first and third quartile value of the data.<br />Determine the value of interquartile range.<br />5. In a survey for a class of final semester student, a group of data was obtained for the number of text books owned.<br /> <br />Number of studentsNumber of text book owned1291115108553210<br /> <br />Find the average number of text book for the class. Use the weighted mean.<br />The following data represent the ages of 15 people buying lift tickets at a ski area.<br />1525261738166021<br />30532840203531<br />Calculate the quartile and interquartile range.<br />A student scores 60 on a mathematics test that has a mean of 54 and a standard deviation of 3, and she scores 80 on a history test with a mean of 75 and a standard deviation of 2. On which test did she perform better?<br />The following table gives the distribution of the share’s price for ABC Company which was listed in BSKL in 2005.<br />Price (RM)Frequency12 – 1415 – 1718 – 2021 – 2324 – 2627 - 2951425763<br />Find the mean, median and mode for this data.<br />

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