Answer to Question #165713 in Statistics and Probability for DAVID

Question

The following were the scores obtained by Form II students in a Mathematics 

49

63

59

58

44

49

51

62

37

30

49

45

52

50

42

54

32

57

41

42

56

44

46

63

44

40

50

46

53

48

37

46

53

68

66

58

36

40

56

37

66

43

40

43

51

59

42

52

46

57

 

 

 

 

 

 

 

 

 

 

 

Required

a)     Group the data into 7 classes and Make a grouped frequency distribution        

  (i)        Mean mark                                                                                                     

(ii)              Mode                                                                                                              

(iii)            Median mark                                                                                                  

(iv)            70^th percentile mark                                                                                       

(v)              Standard deviation                                                                                       

Comment on the distribution of marks 

Answer 1

Frequency Distribution Table

$n = 50$

(i) Mean is calculated by the formula:

$$\bar x=\frac{\sum x_i}{n} = \frac{2452}{50}=49.04$$

(ii) Mode is the value that appears most often in a set and it is 46

(iii) Median is the value separating the higher half from the lower half of a data sample and is calculated by the formula (after sorting the set in ascending order):

$$\frac{n_{25}+n_{26}}{2}=\frac{49+49}{2}=49$$

(iv) 70^th percentile mark:

$n\cdot0.7=50\cdot0.7=35$ it is the index

$n_{35}$ is score 53

thus 70^th percentile mark is 53

(v) Standard deviation is calculated by the formula:

$$s=\sqrt{\frac{\sum (x_i-\bar x)^2}{n-1}}=9.12$$

Comments:

An average score students obtain in mathematics is 49.

The most frequent score among 50 students is 46 (4 students).

The distribution is positively skewed (or right skewed) since mode < median < mean. Wich means that the majority of students obtain scores below the average value of 49.

30% of 50 students obtain scores above 53.

On average, a student's score deviates from the mean by about 9 points.