Statistics - Part I

Statistics is a branch of mathematics that deals with averages, distribution of numbers, etc. of a group of numbers. For example, in cricket, you know that a batsman has scored so many runs in each one day match. You want to know his overall performance in one-day matches. Then how will you find that out? You sum up his individual scores and divide it by the number of one day matches he has played. This will give you the average of his performance in one day matches that he has played.

Statistics creeps into all walks of lives. The sequence of change in signals at a road crossing also depends on statistical inputs. The volume of traffic in either direction, the time of the day, how far is the next signal, etc. determines how long a particular signal will be kept red or green. In Mumbai, for example, the flow of traffic in the morning is more in the southern direction and in the evening, the flow of traffic is more in the northward directions. The traffic signal circuits have to be timed in such a way that such variations during the day are taken into account. The timing of the signal changes will depend on the statistical inputs. In the stock market, the prediction of highs and lows also depends on statistics.

Statistics deals with distribution of numbers in a given set of numbers. The given set of numbers is called the sample. Instead of dealing with single numbers, statistics gives you a handle to deal with an average quantity which is a representative of the set of numbers. Thus statistics deals with averages, standard deviation (from the average), frequency distribution, that is how many times a number occurs in a given set of numbers, etc.

What we will study in this chapter :

1. Mean or average
2. Frequency and frequency table
3. Class interval
4. Frequency polygon

1. Mean or average
Example 1 : Your mother has given you Rs.100 to spend on chocolates and biscuits for a week.  You spend Rs.10 on buying one bar of chocolate in the first week, then you spend Rs.30 on buying a packet of eclairs in the second week. Later on in the third week, you spend Rs.11 on a packet of cream biscuits. What is the average of your spending per week?

                                             Sum of spending                                    Rs. (10 + 30 + 11)
Average   =          =   
                        Number of times a purchase has been made                            3

                          Rs.51
              =                =    Rs.17
                            3

Thus you are spending an average of Rs.17 per week.

Example 2 : In a class of 30 students, the science exam results out of 100 marks are as follows : 10 students get 50 marks, 5 students get 55 marks, 5 students get 65  marks and other 10 students score 75 marks. What is the average percentage of marks the class has obtained in the science exams?

Number of students                  Marks              Total of all marks

           10                                 50                   10 x 50
            5                                  55                    5 x 55
            5                                  65                    5 x 65
            10                                75                    10 x 75
                                                        
            30                                                         1850 
(Total number of student)                                  (Grand total of all marks )

                                                                   Grand total of all marks
Average marks obtained by the class  =     
                                                                   Total number of students

                                                                   1850
                                                     =                       = 61.66
                                                                     30

Thus the class has obtained an average of 61.66  marks in the science exams out of 100 marks..

Definition of mean or average  :

                     Sum of all scores
Mean   =     
                     Number of scores

Scores are the individual values actually occurring at each time. The number of scores is the number of times the same measurement is being done.

Example 3 : The daily temperature recorded in Pune for the past week was : Monday 7ƒC, Tuesday 7.5ƒC,  Wednesday 8ƒC, Thursday 6ƒC, Friday 6.5ƒC, Saturday 7ƒC and Sunday 8.5ƒC. Find the mean temperature recorded in the city of Pune for the past week.

Scores are : 7ƒC, 7.5ƒC, 8ƒC, 6ƒC, 6.5ƒC, 7ƒC and 8.5ƒC

The number of scores : 7

                     Sum of all scores                7ƒC + 7.5ƒC + 8ƒC + 6ƒC + 6.5ƒC + 7ƒC + 8.5ƒC
Mean   =            =   
                     Number of scores                                            7

\ mean =  7.12ƒC

Thus the mean temperature recorded in Pune for the past week is 7.12ƒC.

Example 4 :  Given below is the weekly attendance of a class for a month. Find the mean weekly attendance.

45, 51, 35, 40

In this problem the

scores are : 45, 51, 35, 40 and
the number of score : 4

                     Sum of all scores                     45 + 51 + 35 + 40
Mean   =           =                = 42.75
                     Number of scores                                4

Thus the mean weekly attendance of the class is 42.75. 

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