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Sample Data, Statistics

and the Empirical Rule

Introduction :

Random variable is areal valued function define over a sample space. It says, it is a number

associated with each element in a sample space. Sample data gain from the sample space.

Sample space contains all possible outcomes that occurs in a experiment and we called sample

space a population. When we select few outcomes from the sample space, we called that subset

as sample.

Table 1: Sorted Set of Sample Data

Column1

17.9

21.4

23.6

25.8

26.4

26.9

27.2

27.4

28

28.2

Column2

28.3

28.9

29.1

29.1

29.2

29.3

29.3

30.8

31

31.1

Column3

31.3

31.4

31.6

31.6

32

32.3

32.5

32.8

33.3

33.5

Column4

33.7

33.8

33.9

34

34.7

34.9

34.9

37.6

38.2

38.3

Column5

38.3

38.4

38.6

39

39.5

41.1

Table 2 : Frequency Distribution

Class intervals

15-20

20-25

25-30

30-35

35-40

40-45

Frequency

1

2

14

20

8

1

Graph 1 : Histogram

Frequency

25

20

15

10

5

0

15-20

20-25

25-30

30-35

35-40

Table 3: : Frequency Distribution with Midpoints

Class intervals

15-20

20-25

25-30

30-35

35-40

40-45

Mid point

17.5

22.5

27.5

32.5

37.5

42.5

Frequency

1

2

14

20

8

1

40-45

Table 4 : Summary Statistics

Class

intervals

15-20

20-25

25-30

30-35

35-40

40-45

Mid

point(d)* Frequency(f)

17.5

1

22.5

2

27.5

14

32.5

20

37.5

8

42.5

1

46

Ʃfd = 1440

fd

d-x ̅

17.5

-13.80

45

22.50

385

-3.80

650

1.20

300

6.20

42.5

11.20

1440

Ʃf=46

Ʃ𝑓𝑑

𝑥̅ =

Ʃ𝑓

𝑥̅ =31.304

Ʃ𝑓(𝑑 − 𝑥̅ )2 = 1009.24

Ʃ𝑓(𝑑 − 𝑥̅ )2

𝜎2 =

Ʃ𝑓

𝜎 2 = 21.94

σ = 4.68

46 𝑡𝑒𝑟𝑚

Term of 𝑀𝑒𝑑𝑖𝑎𝑛 = ( 2 )

= 23𝑟𝑑 𝑡𝑒𝑟𝑚

Term of Median = 30-35 interval

𝑛

−𝑐

𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙 + (2

)ℎ

𝑓

Column1

190.44

154.88

202.16

28.8

307.52

125.44

1009.24

cumulative

frequency

1

3

17

37

45

46

ℓ = lower limit of median class interval

C = cumulative frequency preceding to the median class frequency

f = frequency of the class interval to which median belongs

h = width of the class interval

n = sum of frequencies

Median = 26.50

Modal class = 30-35

Mean(𝑥̅ )

Standard deviation(𝜎 2 )

Variance(σ)

Median

Modal class

31.30

21.94

4.68

26.5

30-35

Table 5 : The Empirical Rule

Empirical rule

%

68

95

99.7

Number

of data

31.28

43.7

45.86

Summary :

A population includes all of the elements from a set of data. A sample consists one or more

observations gain by the population. A frequency distribution is an overview of all distinct

values some variable and the number of time they occur. Frequency distribution tells how

frequencies are distributed over values. When making frequency distribution, determine the

range(difference between the highest and lowest observations in data) and decide the number of

classes to estimate approximate size of the interval and then make the class intervals and also

make frequencies according to them.

Histograms are usually presented with vertically rectangular bars with no gap between the

bars. The area under the graph in a histogram represent the total frequency. The X axis gives the

class boundaries and the frequencies is plotted in the Y axis in a histogram. The mid points of

each of the bar in histogram give a bell shaped arc. Since normal distribution is very popular to

model continuous data, histogram may be helpful to understand the data normality.

When discuss the measure of center that measurements, called measures of central tendency.

They are mean, median and mode. Comparison between this three, mean shows as the best

measurement but it also has some disadvantages and median and mode play better roll in

understanding the location of sample data observations. Measures of variation ,called measures

of dispersion. They are standard deviation, range and inter quartile range. Standard deviation is

the most reliable measure. Range is a very poor measure for dispersion. Random number

generator is sometimes used for model sample data with population ones.

The empirical rule is a statistical rule which states that for a normal distribution, almost all

observed data will fall withing three standard deviation of the mean. In particular the empirical

rule predict that 68% of observations fall within the first standard deviation(μ ± σ), 95% within

the first two standard deviations(μ ± 2σ) and 99.7% within the first three standard deviations

(μ ± 3σ).Skewness and symmetry can be identified using the graphical techniques such as

histograms, which clearly shows the shape of a data distributions.

Sample Data, Statistics

and the Empirical Rule

Introduction :

Random variable is areal valued function define over a sample space. It says, it is a number

associated with each element in a sample space. Sample data gain from the sample space.

Sample space contains all possible outcomes that occurs in a experiment and we called sample

space a population. When we select few outcomes from the sample space, we called that subset

as sample.

Table 1: Sorted Set of Sample Data

Column1

17.9

21.4

23.6

25.8

26.4

26.9

27.2

27.4

28

28.2

Column2

28.3

28.9

29.1

29.1

29.2

29.3

29.3

30.8

31

31.1

Column3

31.3

31.4

31.6

31.6

32

32.3

32.5

32.8

33.3

33.5

Column4

33.7

33.8

33.9

34

34.7

34.9

34.9

37.6

38.2

38.3

Column5

38.3

38.4

38.6

39

39.5

41.1

Table 2 : Frequency Distribution

Class intervals

15-20

20-25

25-30

30-35

35-40

40-45

Frequency

1

2

14

20

8

1

Graph 1 : Histogram

Frequency

25

20

15

10

5

0

15-20

20-25

25-30

30-35

35-40

Table 3: : Frequency Distribution with Midpoints

Class intervals

15-20

20-25

25-30

30-35

3...