Arithmetic mean

1. Arithmetic mean Group members Aqsa Ijaz Aqsa Mehmood Ayesha Mumtaz Maira Naseer Sehrish Iqbal Zunaira Munir

2. Arithmetic Mean In mathematics and statistics, the arithmetic mean, or simply the mean or average is the sum of a collection of numbers divided by the total numbers in the collection. OR Arithmetic mean can be defined as:         The sum of all values in the series divided by the total numbers of values in the series.

3. The method of Arithmetic mean is also known as: -                     Arithmetic mean -                     Mean -                     Simple mean -                     Average Methods of arithmetic mean Direct Method (2) Short-Cut Method (3) Step-Deviation • Arithmetic mean for ungroup data(have not been organized in any way , also called raw data) • Arithmetic mean for group data (have been organized into a frequency distribution)

4. Merits: • It is rigidly defined. • It is easy to calculate and simple to follow. • It is based on all the observations. • It is determined for almost every kind of data. • It is finite and not indefinite. • It is readily put to algebraic treatment. • It is least affected by fluctuations of sampling. • Demerits: • The arithmetic mean is highly affected by extreme values. • It cannot average the ratios and percentages properly. • It is not an appropriate average for highly skewed distributions. • It cannot be computed accurately if any item is missing. • The mean sometimes does not coincide with any of the observed value.

5. Arithmetic Mean (ungroup-data) Formula: Mean = sum of elements / number of elements = a1+a2+a3+.....+an/n Arithmetic Mean = Σx/n where x= Individual value n = Total number of values

6. For ungroup data Example: To find the mean of 3,5,7. Step 1: Find the sum of the numbers. 3+5+7 = 15 Step 2: Calculate the total number. there are 3 numbers. Step 3: Finding mean. 15/3 = 5 Ans = 5

7. Arithmetic mean for ungroup data Question no 1 If a baseball pitcher throws three straight strikes to the first batter, two strikes to the second batter, one strike to the third batter, and zero strikes to the fourth batter, what is the average number of strikes the pitcher threw to each of the four batters? Solution: Arithmetic Mean = Σx/n X=3,2,1,0 Σx=3 + 2 + 1 + 0 = 6 N = 4 (since there are four batters)putting values in formula Arithmetic mean=6/4=1.5

8. Question no 2 Find the mean driving speed for 6 different cars on the same highway. 66mph, 57mph, 71mph, 54mph, 69mph, 58mph Solution: Arithmetic Mean = Σx/n N=6 (Total no of cars) Σx=66 + 57 + 71 + 54 + 69 + 58 Σx =375 putting values in formula Arithmetic mean =375/6 Answer: The mean driving speed is 62.5 mph.

9. Arithmetic Mean (group-data) : Formula: Arithmetic Mean = ΣfX/Σf where X = Individual value f = Frequency (number of accurance)

10. For group data Example: X -Value Frequency(f) ΣfX1 2 1 * 2 = 2 2 3 2 * 3 = 6 3 2 3 * 2 = 6 Σf = 7 ΣfX= 14 Step 1: Find Σf. Σf = 7 Step 2: Now, find ΣfX. ΣfX = 14 Step 3: Now, Substitute in the above formula given Arithmetic mean = ΣfX/Σf = 14/7 = 2

11. Classes Frequency of Class Age (years) f 0 --- 5 6 5 --- 10 12 10 ---15 19 15 ---20 3 Find Arithmetic mean for given data. solution: As the formula for A.M (Group data) = ΣfX/Σf So we need “X” first Classes of class Frequency Mid Point Age (years) fX fX 0 --- 5 6 2.5 15 5 --- 10 12 7.5 90 10 ---15 19 12.5 237.5 15 ---20 3 17.5 52.5 Total40 395

12. Question no 4 The following data shows distance covered by 100 persons to perform their routine jobs. Solution: The given distribution belongs to a grouped data and the variable involved is ages of “distance covered”. While the “number of persons” Represent frequencies.

13. Now we will find the Arithmetic Mean as =ΣfX/ Σf =2400/100=24Km

14. Direct Method formula: X=Σx/n or X=ΣfX/Σf Where : n=Total number of values x=Individual value or mid point f=Frequency 2. Short-Cut Method formula: X=A+ΣfD/Σf Where : D=deviation from A i.e D=(x-A) A=assumed mean value 3. Step-Deviation Method formula: X=A+(ΣfU/Σf)*h Where : h=size of class U=step-deviation from A i.e u=x-A/h Methods of arithmetic mean

15. QUESTION 5 The following frequency distribution showing the marks obtained by students in statistics at a certain college. Find the arithmetic mean using (1) Direct Method (2) Short-Cut Method (3) Step-Deviation

17. Direct Method: X=ΣfX/Σf=2745/50=54.9 Marks (2) Short-Cut Method: X=A+ΣfD/Σf Where A=54.5 =54.5+0.4 =54.9 Marks (3) Step-Deviation Method: X=A+(ΣfU/Σf)10 Where A=54.5 =54.5+(2/50)*10 =544.5+0.4 =54.9 Marks

18. Median: -- The median is the middle number. First you arrange the numbers in order from lowest to highest, then you find the middle number by crossing off the numbers until you reach the middle.Example: - use the above data to find the median: 66 74 75 78 82 89\ - as you can see we have two numbers, there is no middle number. What do we do? It is simple; we take the two middle numbers and find the average, ( or mean ). 75 + 78 = 153 153 / 2 = 76.5 Hence, the middle number is 76.5.

19. Mode: -- this is the number that occurs most often.Example: - find the mode of the following data: 78 56 68 92 84 76 74 56 68 66 78 72 66 65 53 61 62 78 84 61 90 87 77 62 88 81 The mode is 78.

20. Mean: -- Also known as the average. The mean is found by adding up all of the given data and dividing by the number of data entries. Example: the grade 10 math class recently had a mathematics test and the grades were as follows: 78 6682897574464464/6=77.3Hence, 77.3 is the mean

21. Arithmetic means: The term between 2 given terms of an arithmetic sequence are also called arithmetic means. Formula: =+(n-1)d 4 arithmetic means between 7and 37 Solution: Formula: +(n-1)d 7,---,---,---,---,37 Here: n=6 Common difference=d=?

22. By formula: +(n-1)d 37=7+(6-1)d 37=7+5d 5d=30 d=6 1: Solution: 7-2=n-1 5+1=n 6=n +(n-1)d -18=-3+(6-1)d -18=-3+5d -15=5d -3=d +(n-1)d -3 =(2-1)(-3) -3=+(1)(-3)

23. -3=+(-3) -3+3= 0 = +2d =0+2(-3) 2:Find the value of x Solution: n =5 +(n-1)d 2x=(x-2)+(5-1)(x-7) 2x=x-2+4(x-7) 2x=x-2+4x-28 2x=5x-30 -3x=-30 X=10

24. Thank you 