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# Statistics

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Statistics
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• 1. T- 1-855-694-8886Email- info@iTutor.comBy iTutor.com
• 2. &#xF0A7; The word &#x2018;statistics&#x2019; appears to have been derived fromthe Latin word &#x2018;status&#x2019; meaning &#x2018;a (political) state&#x2019;.&#xF0A7; In its origin, statistics was simply the collection of dataon different aspects of the life of people, useful to theState.&#xF0A7; Statistics is the study of how to collect, organize, analyze, andinterpret numerical information from data.&#xF0A7; There are two types of statistics:-&#xF0A7; Descriptive statistics&#xF0A7; involves methods of organizing, picturing andsummarizing information from data.&#xF0A7; Inferential statistics&#xF0A7; involves methods of using information from a sample to drawconclusions about the population
• 3. &#xF0A7; Everyday we come across a wide variety of information inthe form of facts, numerical figures, tables, graphs, etc.&#xF0A7; These are provided by newspapers, televisions, magazinesand other means of communication.&#xF0A7; These facts or figures, which are numerical or otherwisecollected with a definite purpose are called data.&#xF0A7; Data is the plural form of the Latin word datum(meaning &#x201C;something given&#x201D;).Categorical or Qualitative Data&#xF0A7; Values that possess names or labels&#xF0A7; Color of M&amp;Ms, breed of dog, etc.Numerical or Quantitative Data&#xF0A7; Values that represent a measurable quantity&#xF0A7; Population, number of M&amp;Ms, number of defective parts, etc
• 4. &#xF0A7; Sampling&#xF0A7; Random&#xF0A7; Involves choosing individuals completely at random from apopulation- for instance putting each student&#x2019;s name in a hatand drawing one at random.&#xF0A7; Systematic&#xF0A7; involve selecting individuals at regular intervals. For instance,choose every 4th name on the roll sheet for your class.&#xF0A7; Stratified&#xF0A7; Stratified sampling makes sure you&#x2019;re equally representingcertain subgroups: for instance, randomly choose 2 males and2 females in your class .&#xF0A7; Cluster&#xF0A7; Cluster sampling involves picking a few areas and samplingeveryone in those areas. For instance, sample everyone in thefirst row and everyone in the third row, but no one else.
• 5. &#xF0A7; Raw data &#x2013; Data in the original form.&#xF0A7; Example - the marks obtained by 7 students in amathematics test :55 36 95 73 60 42 25&#xF0A7; Range : The difference between the lowest andhighest values.In {4, 6, 9, 3, 7} the lowestvalue is 3, and the highest is9, so the range is 9-3 = 6.&#xF0A7; Convenience&#xF0A7; A convenience sample follows none of these rules in particular:for instance, ask a few of your friends.
• 6. &#xF0A7; Frequency&#xF0A7; Consider the marks obtained (out of 100 marks) by 21 studentsof grade IX of a school:92 95 50 56 60 70 92 88 80 70 72 70 92 5050 56 60 70 60 60 88&#xF0A7; Recall that the number of students who have obtained a certainnumber of marks is called the frequency of those marks.&#xF0A7; For instance, 4 students got 70 marks. So the frequency of 70marks is 4.&#xF0A7; To make the data more easily understandable, we write it in atable, as given below:
• 7. MarksNumber of students(i.e., the frequency)50 356 260 470 472 180 188 292 395 1Total 21Table is called an ungrouped frequency distribution table,or simply a frequency distribution table
• 8. &#xF0A7; Frequency distribution table consists of variouscomponents.&#xF0A7; Classes&#xF0A7; To present a large amount of data so that a reader can makesense of it easily, we condense it into groups like 10 - 20,20 - 30, . . ., 90-100 (since our data is from 10 to 100).&#xF0A7; These groupings are called &#x2018;classes&#x2019; or &#x2018;class-intervals&#x2019;.&#xF0A7; Class Limits:&#xF0A7; The smallest and largest values in each class of a frequencydistribution table are known as class limits. If class is 20 &#x2013; 30then the lower class limit is 20 and upper class limit is 30.&#xF0A7; Class Size&#xF0A7; their size is called the class-size or class width, which is 10 inabove case.
• 9. &#xF0A7; Class limit&#xF0A7; Middle value of class interval also called Mid value.&#xF0A7; If the class is 10 &#x2013; 20 thenclass limit&#xF0A7; Class frequency:&#xF0A7; The number of observation falling within a classinterval is called class frequency of that class interval.2limlim itHigheritLower1522010
• 10. &#xF0A7; Consider the marks obtained (out of 100 marks) by 100students of Class IX of a school&#xF0A7; Class = we condense it into groups like 20-29, 30-39, . . .,, 90-9995 67 28 32 65 65 69 33 98 967 6 42 32 38 42 40 40 69 95 9275 83 76 83 85 62 37 65 63 4289 65 73 81 49 52 64 76 83 9293 68 52 79 81 83 59 82 75 8286 90 44 62 31 36 38 42 39 8387 56 58 23 35 76 83 85 30 6869 83 86 43 45 39 83 7 5 66 8392 75 89 66 91 27 88 89 93 4253 69 90 55 66 49 52 83 34 36
• 11. Recall that using tally marks, the data above can be condensed intabular form as follows:
• 12. &#xF0A7; Frequency Distribution Graph&#xF0A7; Histogram&#xF0A7; Frequency Polygons&#xF0A7; Categorical data graph&#xF0A7; Bar Chart&#xF0A7; Pie Chart
• 13. &#xF0A7; Bar Chart&#xF0A7; It is a pictorial representation of data in which usuallybars of uniform width are drawn with equal spacingbetween them on one axis (say, the x-axis), depictingthe variable.&#xF0A7; The values of the variable are shown on the other axis(say, the y-axis) and the heights of the bars depend onthe values of the variable.&#xF0A7; For the construction of bar graphs, we go throughthe following steps :&#xF0A7; Step 1 : We take a graph paper and draw two linesperpendicular to each other and call them horizontal andvertical axes.&#xF0A7; Step 2 : Along the horizontal axis, we take the values ofthe variables and along the vertical axis, we take thefrequencies.
• 14. &#xF0A7; Step 4 : Choose a suitablescale to determine the heightsof the bars. The scale is chosenaccording to the spaceavailable.&#xF0A7; Step 5 : Calculate the heightsof the bars, according to thescale chosen and draw the bars.&#xF0A7; Step 6 : Mark the axes withproper labeling.&#xF0A7; Step 3 : Along the horizontal axis, we choose the uniform(equal) width of bars and the uniform gap between the bars,according to the space available.
• 15. &#xF0A7; Histogram&#xF0A7; This is a form of representationlike the bar graph, but it is usedfor continuous class intervals.&#xF0A7; It is a graph, including verticalrectangles, with no space betweenthe rectangles.&#xF0A7; The class-intervals are takenalong the horizontal axis and therespective class frequencies on thevertical axis using suitable scaleson each axis.&#xF0A7; For each class, a rectangle is drawn with base as width ofthe class and height as the class frequency.
• 16. &#xF0A7; Frequency Polygons&#xF0A7; A frequency polygon is thejoin of the mid-points ofthe tops of the adjoiningrectangles.&#xF0A7; The mid-points of the firstand the last classes arejoined to the mid-points ofthe classes preceding andsucceeding respectively atzero frequency to completethe polygon.&#xF0A7; Frequency polygons can also be drawn independentlywithout drawing histograms.&#xF0A7; For this, we require the mid-points of the class-intervalsused in the data.
• 17. &#xF0A7; Frequency Polygons&#xF0A7; Frequency polygons are used when the data iscontinuous and very large.&#xF0A7; It is very useful for comparing two different sets of dataof the same nature
• 18. &#xF0A7; Pie Chart&#xF0A7; Pie chart, consists of a circular region partitioned intodisjoint sections, with each section representing a part orpercentage of a whole.&#xF0A7; To construct a pie chart firstly we convert the distributioninto a percentage distribution.&#xF0A7; Then, since a complete circle corresponds to 3600 , weobtain the central angles of the various sectors bymultiplying the percentages by 3.6.42%25%20%13%Sales1st Qtr2nd Qtr3rd Qtr4th Qtr
• 19. &#xF0A7; The term central tendency refers to the "middle" value orperhaps a typical value of the data, and is measured usingthe mean, median, or mode.&#xF0A7; Each of these measures is calculated differently, and the onethat is best to use depends upon the situation.Mean The averageMedianThe number or average of the numbersin the middleMode The number that occurs most
• 20. Mean&#xF0A7; The mean(or average) of a number of observationsis the sum of the values of all the observations dividedby the total number of observations.&#xF0A7; It is denoted by the symbol , read as &#x2018;x bar&#x2019;xxnnsobservatioofnumberTotalnsobservatiotheallofSumxmeanThex
• 21. Median&#xF0A7; The median is that value of the given number ofobservations, which divides it into exactly two parts.&#xF0A7; So, when the data is arranged in ascending (ordescending) order the median of ungrouped data iscalculated as follows:&#xF0A7; When the number of observations (n) is odd,&#xF0A7; The median is the value of the Observation .thn21Median
• 22. Median is their meanMedian:&#xF0A7; When the number of observations (n) is even,&#xF0A7; The median is the mean of the andobservation .thn2thn12Mode&#xF0A7; The Mode refers to the number that occurs the mostfrequently.&#xF0A7; Multiple modes are possible: bimodal or multimodal.
• 23. Example&#xF0A7; Find the mean, median and mode for the followingdata: 5, 15, 10, 15, 5, 10, 10, 20, 25, 15.(You will need to organize the data.)5, 5, 10, 10, 10, 15, 15, 15, 20, 25&#xF0A7; Mean:&#xF0A7; Median: 5, 5, 10, 10, 10, 15, 15, 15, 20, 25 Listing thedata in order is the easiest way to find the median.&#xF0A7; The numbers 10 and 15 both fall in the middle. Averagethese two numbers to get the median.5.1221510
• 24. &#xF0A7; Mode&#xF0A7; Two numbers appear most often: 10 and 15.&#xF0A7; There are three 10s and three 15s.&#xF0A7; In this example there are two answers for the mode.Call us for moreinformationwww.iTutor.com1-855-694-8886Visit