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Statistics Formula Business, Study notes of Statistics

Manipal Academy of Higher EducationStatistics

Formulan statistics. Business Statistics sa

Typology: Study notes

2022/2023

Uploaded on 10/27/2023

sandeep-13
sandeep-13 🇮🇳

2 documents

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bg1
The important statistics formulas are listed
in the chart below:
Mean
Median
Mode
Variance
Standard
Deviation
If nis odd, then
M=
()th
term
If nis even, then
M=
(4)lterm+(+1)"term
2
The value which
OCCurs most
frequently
S=o=
S(-)'
X=
Observations
given
n= Total
number
of observations
n= Total
number
of observations
X=
Observations
given
=Mean
n= Total
number
of observations
X=
Observations
given
=Mean
n=Total
number
of observations
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

Partial preview of the text

Download Statistics Formula Business and more Study notes Statistics in PDF only on Docsity!

The important statistics formulas are^ listed

in the chart below:

Mean Median Mode Variance Standard Deviation If n (^) is odd, (^) then M= () th

term

If n^ is even, then M= (4)lterm+(+1)"term 2 The value which OCCurs (^) most frequently S=o=

S(-)'

X= Observations given

n= Total

number of (^) observations

n= Total

number of (^) observations X= Observations given = (^) Mean n= Total number of (^) observations X= Observations given = (^) Mean n=Total number of (^) observations

Mean Deviation Formula The mean^ deviation is^ also known as^ the mean (^) absolute deviation and is (^) defined as the mean^ of the absolute deviations of the observations from (^) the suitable average

which may^ be the arithmetic mean,^ the

median or^ the mode.

The formula to^ calculate Mean deviation is

as (^) stated beloW: Mean Deviation (^) from Mean = Mean Deviation (^) from Median = Here,

represents the summation.

X (^) =Observations X = (^) Mean N= The number of^ observations

M= Median

For (^) frequency distribution, (^) the mean

deviation is^ given by:

M. (^) D = IM Lix-X

Solved (^) Example

Question: Find the mean^ absolute deviation

ofthe following data set:

26, 46, 56, 45, 19,^ 22, (^) 24. Solution:

Given set of data is:

26, 46, 56, 45, 19,^ 22, (^24) Mean =^ (26 +^46 +^56 +^45 +^19 +^22 +^ 24)/ =

i.e.

Now construct the following^ table for MAD:

26 46 56 45 19 22 24

  • 12 7 22

(8-+12+22+11+15+12-+10) 12

22 15

Now, let's find out the average^ of all the

absolute values: 12 10

Therefore, the mean^ absolute deviation of

the given^ data set^ is^ 12.857.

Population Mean Formula The ratio wherein the addition of the values to (^) the (^) number of (^) the value is^ a^ population mean -^ if^ the possibilities are (^) equal. A

population mean^ include each element

from the set^ of^ observations that can^ be

made.

The population mean^ can^ be found using

the following^ formula: Where, = (^) Sum of the values N= Number of^ the value

Solved Example

Solution:

Question: Find^ the population mean^ of^ the

following (^) numbers 1,^ 2,3,4,^ 5. Given, -1,2, (^) 3,4, =|+2 (^) +3+ 4 +5= N= Population Mean =

l=

N

Population Mean^ =

X

The (^) sample (^) size (^) formula helps us (^) find (^) the

accurate sample size through the difference

between the population and the sample. To

recall,the number of observation in a^ given

sample population is known as sample size.

Since it^ not^ possible (^) to survey (^) the whole population, we^ take a^ sample from (^) the

population and then conduct a survey or

research. The sample size is denoted by "n"

or "N", Here, it is written as "Ss".

Learn (^) More: (^) Confidence Interval (^) Formula Sample Size^ Formula for Infinite (^) and Finite Population

We should know that the sample size

that

we are taking from

the population,^ will^ not

hold good (^) for (^) the whole sample. We (^) have a

level of confidence and margin of error to

calculate that the sample size is dccurate or

not.Confidence level (^) helps describe how

sure youare

that the results of the survey

hold true or^ aCcurate.

The sample size for an^ infinite (unknown)

population and for^ a^ finite^ (known)

population is^ given as: Formulas (^) for (^) Sample Size (^) (Ss) For (^) Infinite (^) Sample (^) Size For (^) Finite (^) Sample Size Where,

  • (^) SS= Sample size
  • (^) Z= Given (^) Zvalue ss =^ (2²p (^) (-p)]/ c
  • (^) Pop= Population

ss/ [1+^ {(ss- 1)/Pop}l

p=Percentage of (^) population •C= Confidence^ level

Sample Size^ Formula Example Question: Find^ the sample^ size for^ a^ finite and infinite^ population^ when^ the percentage (^) of 4300 population is^ 5, confidence level^99 and confidence interval is 0.01? Solution: Z= From^ the z-table,^ we have the value of confidence level,^ that is^ 2.58 by applying given (^) data in (^) the (^) formul: SS = (2.58)x0.05x(1-0.05) = (^316) Sample (^) size for (^) finite population

= (^294) New SS^ = 294

Quartile (^) Formula A (^) quartile (^) divides (^) the set of (^) observation into (^4) equal parts. The (^) middle term, (^) between (^) the median and first term^ is^ known as^ the first or Lower (^) Quartile (^) and is (^) written as^ Q1.^ Similarly, the value of^ mid term^ that lies between the last term and the median is^ known as^ the third or^ upper^ quartile and is^ denoted as^ Q3. Second Quartile^ is^ the median and is^ written as (^) Q2. When (^) the set^ of^ observation is^ arranged in an (^) ascending (^) order, then the 25th percentile is (^) given as:

Q1=

n

4

Qs=

th The (^) second quartile or^ the 50th percentile or the Median^ is^ given^ as: th 4

Term

The third Quartile of (^) the 75th (^) Percentile (Q3) is (^) given as:

Term

th

Term

The Upper quartile is^ given by rounding to the nearest^ whole^ integer^ ifthe solution^ is coming in^ decimal number. The^ major use of (^) the lower (^) and upper^ quartile helps is^ that it (^) helps us^ measure^ the dispersion in (^) the set of (^) the (^) datagiven. The dispersion is^ also called "inter^ quartile^ range",^ denoted as^ IQR, inter quartile^ range^ is^ the difference between lower^ and upper^ quartile. IQR =^ Upper^ Quartile -^ Lower Qu To find (^) the quartile we^ first (^) need to arrange the values in^ ascending^ order.^ Then^ we need to^ put^ the formula to^ use.^ Let's^ solve one (^) example to (^) make it (^) clear to you:

Solved example Question: Find^ the median, lower quartile, upper (^) quartile (^) and inter-quartile range (^) of the following^ data set^ of^ scores:^ 19,^ 21,^ 23,^ 20,

Solution:

First, (^) lets arrange^ of (^) the values in an

ascending order: 19,^ 20,21,^ 23, 23,^ 24, 25,^ 27,

31 Now (^) let's (^) calculate the Median, 2= ("Term Q2= (^) ("Term = (^) 5h Term = (^23)

Lower Quartile:

1=()"Term

() Term^

= 2.5th Term

Upper Quartile:

th Q =())"Term Q 4 4 3(9+1) th

30 7.5th Term

Term

=

Average (^) of (^) 2nd (^) and 3rd terms

(20 +^ 21)/

=

=

Lower Quartile

Average (^) of 7th (^) and 8th terms

(25 +^ 27)/

=

=

Upper Quartile

IQR =^ Upper quartile -^ Lower (^) quartile

  • (^) 20.

Example 2:^ A^ student wrote^2 quizzes. In^ the

first quiz,^ he scored 80 and in^ other,^ he

scored 75. The mean^ and standard

deviation of^ first^ quiz^ are70^ and 15

respectively, while the mean^ and standard

deviation of^ the second quiz^ are^54 and 12

respectively. The results follow the normal

distribution. What can^ you^ conclude about

the student's result by^ seeing^ their z

SCores? Solution:

Calculation of^ student's Z^ score^ for first quiz:

Standardized random variable X=^80

Mean, =

Population standard deviation

= (^15) Formula for^ Z^ score^ is^ given^ below: Z

Score=(x-z)/o

=

= (^) 0. Calculation of^ student's Z^ score^ for^ second quiz:

Standardized random variable x=

Mean (^) %= 54

Population standard deviation

= (^12) Formula for Z^ score^ is^ given^ below: Z (^) Score =^ (x

-)/o

= (^175)

Since Z^ score^ of^ second quiz^ is^ better than

that of^ first^ quiz,^ hence^ it^ is^ concluded^ that

he did^ better in^ second quiz.

The CentralLinit Theorem is^ the sampling

distribution of^ the sampling means^ approaches a

normal distribution^ as^ the sample size^ gets^ larger,

no matter what the shape of the data distribution.

An (^) essential component of (^) the Central Limit

Theorem is^ the average^ of^ sample^ means^ willbe

the population mean. Similarly, if^ you^ find (^) the average^ of all of (^) the standarddeviations^ in^

your sample, youwill

find (^) the actual (^) standarddeviation (^) for your

population.

  • (^) Mean of sample is sanme (^) as

the mean^ of^ the

population.

The standard deviation of the sample is

equal to^ the standard deviation of^ the

populationdivided^ by^ the square^ root^ of^ the

sample size. Central limit^ theorem is^ applicable^ for^ a

sufficiently large sample sizes (n 2 30). The

formula for^ central limit^ theorem can^ be

stated as^ follows:

and

Sl

Where,

P= Population^ mean

g= (^) Population (^) standard deviation

Sample mean = (^) Sample standard deviation

n= Sample size

Linear Regression Formula Linear regression is (^) the most (^) basic and commonly (^) used predictive analysis. One variable is^ considered to^ be an^ explanatory variable, (^) and the other is^ considered to^ be a dependent variable. For^ example, a^ modeler might want to (^) relate the weights of individuals to^ their heights using a^ linear regression model. There are^ several linear (^) regression analyses available to^ the researcher. Simple linear regression

  • (^) One dependent variable (interval or (^) ratio)
  • (^) One (^) independent variable (interval or (^) ratio or (^) dichotomous) Multiple linear regression
  • (^) One (^) dependent variable (interval or (^) ratio) Two or^ more^ independent variables (interval or ratio or (^) dichotomous) Logistic regression
  • (^) One (^) dependent variable (binary)
  • (^) Two or (^) more (^) independent variable(s) (interval or^ ratio or^ dichotomnous) Ordinal regression One (^) dependent variable (ordinal) One or^ more^ independent variable(s) (nominal or^ dichotomous) Multinomial (^) regression
  • (^) One (^) dependent variable (nominal)
  • (^) One or (^) more (^) independent variable(s) (interval or^ ratio or^ dichotomous) Discriminant (^) analysis
  • (^) One dependent variable (nominal) One or^ more^ independent variable(s) (interval or ratio) Formula for linear regression equation is given by:

y=a+ bæ

a (^) and (^) bare given by the following formulas: a(intercept) =

b (slope)=2y-(=)()

Where, X (^) and yare two (^) variables on (^) the regression line. b= Slope^ of^ the^ line. a=rintercept of the line. X=Values of^ the first (^) data set. y=Values of^ the second data set.

Solved Examples

Question: Find^ linear regression (^) equation for the following^ two^ sets^ of^ data: Solution: y Construct (^) the following^ table: X a (^) = 2 a = 4 6 8 = (^20) 4x144-20x 4x 120-

ny-(z\D)

b=0. 2 g=1. 3

y=a +^ bx

7 5 = 10 n(- 25x120-20x 144 4(120)- y=1.

  • (^) 0.95 x 4 7 y 4 16 36 64 Linear (^) regression is^ given by: = (^120) 6 5 6 8 10 28 30 80 æy = (^144)