Gait Learig Guides Itermediate Statistics Data processig & display, Cetral tedecy Author: Raghu M.D.
STATISTICS DATA PROCESSING AND DISPLAY Statistics is the study of data or umerical facts of differet groups. Data may be the marks obtaied by studets from differet schools or umber of childre i each family i a group of houses ad ay other quatifiable fact. Data is a set of differet values take by variable characters quatifyig the above metioed umerical facts. Data which have ot bee grouped is called raw data. Whe the raw data is grouped, for example marks obtaied by school studets from rural area, it ca provide more useful iformatio. Display of grouped data is doe through groups, tables ad charts. Example : A survey of 0 apartmets i a block revealed the followig data about the umber of childre per household. House No. 5 6 7 8 9 0 No. of childre 0 5 Aswer: Followig table give the data rearraged i the form of a ascedig order House No. 0 5 7 8 6 9 No. of childre 0 5 table draw below gives the tally chart of umber of families havig to 5 childre ad total of tally. Distributio No of Childre 0 5 Tally chart No. of Families Example : Percetage marks scored by 0 studets i mathematics are as follows. Prepare a frequecy distributio table. Select a suitable class iterval, fid the class size ad class marks for the iterval chart. Itermediate-Statistics of 8 0 MDR www.learigforkowledge.com/glg
6 60 55 5 6 65 50 7 5 8 60 57 8 55 60 5 55 5 5 5 7 5 5 60 5 5 Aswer: Miimum value of array = Maximum value of array = 65 Differece is Hece recommeded class itervals are 0-5, 5-50, 55-60, 60-65. Class size is 5. The differece betwee two mid poits of class itervals class marks are the ceters of class itervals. They are.5, 7.5, 5.5, 57.5, 6.5 Class Iterval 0-5 5-50 50-55 55-60 60-65 Tally 5 6 Example : distributio of commutig time i mi its for class itervals is give below. Draw the cumulative frequecy table. Note: Cumulative frequecy correspodig to a class iterval is the sum of all frequecies icludig ad up to the class iterval. Table Class Iterval 0-8 8-6 6- - -0 0-8 Aswer: 8 6 5 7 Cumulative frequecy Cumulative frequecy table Class Iterval 0-8 8-6 6-8 6 Cumulative frequecy 8 0 6 60 75 Remarks 0 + 8 8 + 0 + 6 6 + 60 +5 - DATA PRESENTATION -0 5 0-8 7 8 75 + 7 Graphical presetatio of data makes them more oticeable ad easily uderstood. The three importat types of charts are Histogram, polygo ad Cumulative frequecy graphs. Itermediate-Statistics of 8 0 MDR www.learigforkowledge.com/glg
I a histogram class itervals are o the x axis ad correspodig frequecies o the y axis. Data is preseted as a series of adjacet rectagles. Where as, i a frequecy polygo mid. poits of class itervals are marked o the X axis ad correspodig frequecies are marked o the Y axis. Poits correspodig to the x ad y coordiates are marked ad joied by lies to show the frequecy polygo. I the case of cumulative frequecy diagram the y coordiates correspod to the cumulative frequecies. Data ca be i the iclusive form (0-0, 0-0, - - - - - Class Iterval) or exclusive form (0-9, 0-9 - - - - - - Class Iterval). Exclusive form ca be coverted to iclusive form (0.5 to 9.5, 9.5 to 9.5 - - - - - Class Iterval) ad diagrams plotted. Class iterval Mid poit 0-5 5-0 8 0-5 8 5-0 8 0 0-5 5-0 8 Aswer: Features of Histograms X - axis = class iterval Y - axis = frequecy Data represeted as a set of adjacet rectagles with o gaps Histogram 0 5 0 5 0 5 0 Class Iterval Example : Draw a frequecy polygo ad a cumulative frequecy graph for the followig data. Class Iterval Mid poit Cumulative 0-8 8 8 8-6 6-0 6 8-8 6 Itermediate-Statistics of 8 0 MDR www.learigforkowledge.com/glg
-0 6 5 77 0-8 7 8 CUMULATIVE FREQENCY DIAGRAM 80 70 ( (6.77) (.8) Cumulative 60 50 0 (6.8) (.6) NOTE: Start the Cumulative frequecy diagram form (0.0) a startig value is specified 0 0 (.) 0 (.7) (0, 0) 8 6 0 8 Class Iterval FREQUENCY POLYGON 60 50 0 0 0 0 (.) (0.6) (8.) (6.5) (.7) (.7) NOTE: complete the polygo by cosiderig two imagied class itervals (0 to 8) ad (8 to 56). Assume the frequecies will reach 0 at the mid poits of these class itervals (5.0) (0,-) 0 8 6 0 8 Class iterval Itermediate-Statistics of 8 0 MDR www.learigforkowledge.com/glg
CENTRAL TENDENCY Cetral tedecy comprises of three importat statistical parameters, mea, mode ad media. Together with lower ad upper limits they provide a reasoably complete picture of data from a group. Mea or arithmetic is the average of umbers i the list of data. Mode is the most commo umber ad media is the umber at the ceter of a data list arraged i ascedig or descedig order. x x x x - - - -x Mea where x, x - - - -x are values of umerical data Also mea of (x a), (x a) - - - -(x x a where a is a costat a) Example 5: Followig list comprises of salaries of six employees Employee No. Salary Rs. 5 6 7 800 00 000 00 000 00 (x) Fid the mea, mode ad media of the salaries. If a seveth employee jois the ew mea will be Rs.050. Fid the salary of the 7 th employee Aswer: Arrage the list i the ascedig order 800, 000, 000, 00, 00, 00 x x - - - - - - x,600 mea x 6,600 6 Rs. 00 Mode or the most commo umber is Rs.000/- which appears twice ad the rest appear oly oce. Media is the middle umber Rs. 00/- th (6 beig a eve umber the media value correspod to, or umber) New mea correspodig to 7 employees = Rs. 050 7 050 - x 6 00 where x is the7 salary of the7 x 7 050-6 00 or x Rs.750. th employee Itermediate-Statistics 5 of 8 0 MDR www.learigforkowledge.com/glg
EXERCISES. Explai the followig terms:. Data.. Histogram. Mea 5. Mode. The umber of workers each day for 6 days at a buildig site ad give below. 6, 7,, 5, 0,, 7,,, 9,,,, 0, 5, 5, 5 5, 7, 5, 5, 6,, 7,, 9, 6, 6,, 5, 6,,, 7,,,,, 9, 0, a. Arrage the data as a ascedig order b. Draw a tally chart ad frequecy table c. Select suitable class itervals. Followig table gives heights of studets i a class. Complete the cumulative frequecy colum Height (cm) 50-55 55-60 60-65 65-70 70-75 75-80. A fisherma weighted a sample of his catch ad obtaied the followig data. Draw a histogram showig the frequecy distributio of his catch. Weight 0-5 5-0 0-5 5-0 0-5 5-0 5 6 5. The umber of daily absetees over a term i a school is listed below. Draw the frequecy polygo ad cumulative frequecy diagram. Draw the followig polygo ad cumulative frequecy diagram. No. of absetees 0-5 5-0 0-5 5-0 0-5 5-0 0-5 Cumulative frequecy 7 6 0 6 Itermediate-Statistics 6 of 8 0 MDR www.learigforkowledge.com/glg
6. Fid the media of the followig lots of umbers a.,, b.,,, c. 70, 7, 77, 80, 85 d.,,,, 5, 6, 0,,, 7 e. Whole umbers 0 to 6 iclusive of both 0 ad 6 7. Fid the mea ad mode of the followig sets of scores. a), 0,, 7,, 8,,, 6 b), 6, 5,,,, 5,,,,, 9, 0 c) 5, 0, 5, 0, 5, 0, 5, 50, 5, 0, 5, 0 8. Mea of umbers is if 5 is added to each umber fid the ew mea. Write dow a set of 5 umbers which ca give a mea of. 9. I a sciece test, out of 0 marks studets scored 5, studets marks, studets 7 marks ad studet scored marks. Fid the mea, media ad mode of test scores. 0. Cumulative frequecy table of exam marks obtaied by studet is give below. Use this iformatio to draw a frequecy table ad a frequecy polygo. No. of studets 7 5 9 5 5 5 Below 0 marks 0 marks 0 marks 0 marks 50 marks 60 marks 70 marks 80 marks 90 marks 00 marks Itermediate-Statistics 7 of 8 0 MDR www.learigforkowledge.com/glg