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Math and Statistics Business, Cheat Sheet of Statistics

Manipal Academy of Higher EducationStatistics

Math Math Statistics Business Statistics

Typology: Cheat Sheet

2022/2023

Uploaded on 10/27/2023

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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, thnen
M=
(G)titerm-+(+1)term
2
The value which
OCCurs most
frequently
-a)?
S(-)?
S=oVn
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
pf12
pf13
pf14
pf15
pf16
pf17
pf18

Partial preview of the text

Download Math and Statistics Business and more Cheat Sheet 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, thnen

M=

(G)titerm-+(+1)term

2

The (^) value which

OCCurs (^) most

frequently

-a)?

S(-)?

S=oVn

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

0O

300+Centres

I 140+ Cities

Mean Deviation

Formula

Mean Deviation

Formula

Click to chat on^ Whatsapp 9

The

mean

deviation is^ lso known

as

the

mean absolute deviation and is defined as

the

mean

of the absolute deviations of the

observations from^ the suitable^

average

which

mnay

be the arithmetic

mean,

the

median

or the mode.

The formula to calculate Mean deviation is

as

stated below:

Mean Deviation from Mean

Here,

Mean Deviation (^) from Median

represents (^) the summation.

X =Observations

X

= Mean

N= The number of observations

M= Median

S Vo)^

4G

For (^) frequency distribution,

the

mean

deviation is given by:

LTE

M. D

==

M.

Díahut

median)

=

When the

mean

deviation is^ calculated

about the median,^ the formula becomes:

M

Mean Absolute

Deviation Fornmula

Average absolute deviation of the collected

data set is^ the

average of absolute

deviations (^) froma

centre (^) point of the (^) data

set. (^) Abbreviated as^ MAD,^ Mean absolute

deviation has four^

types (^) of deviations (^) that

are derived

by central

tendency, mean

median and mode and standard deviation.

Mean (^) absolute deviation is,^ however, (^) best

used

as it is more^ accurate and

easy (^) to use

in real-life situations.

The formula for Mean Absolute Deviation

(MAD) (^) is as (^) follows:

Where

X;= (^) Input data values

= Mean value for a^ given^ set of data,

MAD

n= (^) Number of^ data values

To (^) find MAD, you need

to (^) follow below steps:

Calculate the

mean for the

given set of^ data.

Find the difference between each value

present (^) in the data set and the

mean that

gives you^ the absolute value.

  • Find (^) the average^ of all (^) the absolute values of

the difference between the data set and the

mean that

gives the

mean absolute

deviation

(MAD).

Solved Example

Question: Find the

mean

absolute deviation

of the

following data

set:

26, (^) 46, 56, (^) 45,19, 22,

Solution:

1.e

Given set of data is:

26, 46, 56, 45,

19, (^) 22, 24

= 34

Mean

=

(

19 +^ 22

24)/

=

238/

= 34

Now construct the following^ table for MAD:

26 -8^8

Mean Absolute

Deviation Formula

Average absolute deviation^

of the collected

data

set is the

average of absolute

deviations froma

centre point^ of the data

set. Abbreviated

as MAD, Mean (^) absolute

deviation has four^

types of deviations (^) that

are derived by^ central tendency,^

mean

median and mode and standard deviation.

Mean absolute deviation^

is,

however,

best

used

as it is more^ accurate and

easy (^) to use

in real-life situations.

The formula for Mean Absolute Deviation

(MAD) (^) is as

follows:

Where

MAD =)

i-

X;= (^) Input data values

=

Mean value for

a (^) given set of data,

n=

Number of^

data

values

To find

MAD,

you

need

to (^) follow below steps:

Calculate the

mean

for

the

given set of

data.

Find the difference between each value

present in

the data

set

and the

mean

that

gives

you the (^) absolute

value.

Find the

average of all the absolute values of

the difference between the data set^ and the

mean

that

gives

the

mean

absolute

deviation

(MAD).

Population

Mean

Formula

The ratio wherein the addition of^ the values

to

the number

of

the value

is

a

population

mean

ifthe

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

followingnumbers

1, 2, 3,4, 5.

Given,

X;

3, 4, 5

X;

  • 4

5

= 15

N=

Population Mean

N

15

5

u=

N

Population

Mean

X;

Margin of^

Error

Formula

The margin^ of^

error

is a^ statistic

expressing

an

amount of (^) random sampling

error

in

a

survey's results.

It

asserts a

likelihood

that the result^ from^

a

sample is

close

to

the number

one

would get^

if the

whole population (^) had been queried.

In

simple words,

the

margin of

error

is the

product of critical value

and the standard

deviation.

The margin^ of^

error is

denoted

byE and the

formula is given as,

The margin of Error

= z^ x

where,

n=

sample size

o=

Population

Standard Deviation

Z =Z SCore

Solved (^) Examples

Question:

A

random sample^

of 30

students

has

average yearly earnings of^2450 and

a

standard deviation^

of 587. Find (^) the margin

of

error if c

=

0.95?

Solution:

Given

n=30,

Standard Deviation=^587

o

= 587

C= 0.

At (^) 95%

level

of

confidence

z =

Margin of^

error

= z x

o//n

=

x

587/V

=

x

=

  • 210

(approx)

Sample

Size

Formula

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 Fornmula^ 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^ accurate or

not. Confidence level^ helps^ describe how

sure you are 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

ss

= (2²p

(-p)l/

c

For Finite (^) Sample Size

Where,

  • SS= Sample^ size

Z= (^) Given

Z

value

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

Pop

= Population

ss/ [1+{(ss^

  • 1)/Pop}l

Check:

Z Score Table

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?

Sample Size^

Fornmula

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

formula:

SS(2.58)

x0.05x

(1-0.05)

Sample size for finite population

14316–I

4300

New

SS

Quartile Formula

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

termn 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:

is given as:

The second quartile

or

the 50th percentile^

or

the Median^

is given

as:

Q

n+

th

3(n

The third Quartile of the 75th Percentile (Q3)

Term

th

Term

th

Term

The (^) Upper quartile is (^) given by rounding to

the nearest^ whole integer^

if

the 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

data

given. 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, (^) 27,25, 24, 31?

Solution:

First, lets

arrange

of

the values

in an

ascending order:

31

Now (^) let's calculate the

Median,

Q2= ()"Term

Q2=()"Term

th

=5th Term

=

Lower Quartile:

Q1=("Term

=

()*

Term 2.5th^ Term

Upper

QUartile:

30

Qs=()

Term

th

th

Term

7.5th Term

Average of (^) 2nd and

3rd terms

(

Lower

Quartile

Average of 7th and 8th^

terms

(

27)/

Upper (^) Quartile

IQR

Upper quartile

Lower (^) quartile

Z

Score Formula

Z-scores

are (^) expressed in terms of

standard

deviations from their

means. Resultantly,

these

z-scores

have

a

distribution with^

a

mean of 0 and a standard deviation of l.^ The

formula for^ calculating^ the standard

score

is given

below:

Formula for^

Z

Score

Where,

•X= Standardized (^) random variable

=

Mean

•g= Standard deviation.

Also Check: Z-Score Table

Solved Examples

z Score

=

(x-* )/a

Following are^

the few^

problems

based

on Z

sCore:

Example

1: How (^) well did Ram perform^ in her

English (^) coursework compared to

the other

(^50) students?

To answer this question,^

can

be re-phrase^

it

as

What percentage^ (or^ number) of

students scored

higher

than Ram^ and what

percentage (or number) of^ students scored

lower than Ram?

First,

let's reiterate that

Ram (^) scored 70 out^ of l00, (^) the

mean score

was

60, and the

standard deviation^

was 15.

English

Coursework

= (

15

= 10/

=

Score

Z Score

= (x

)/o

(x)

70

Mean

(3)

60

Standard

Deviation (o)

15

The z-score^ is 0.67 (^) (to 2 decimal places),

but

now we need to^ work out the

percentage (or number)

of students that

scored

higher and lower^ than Ram.

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

xX

Mean, X=

Population

standard deviation^

15

Formula for^

Z score

is given below:

Z

Score

=

(x-)/o

=

(80-

Calculation of student's

Z score

for second

quiz:

Standardized random variablex= 75

Mean

X= 54

Population (^) standard deviation

  • 12

Formula for^

Z score

is given below:

Z

Score

=

(x-)/o

=

(75-54)

=

Since

Z score

of

second

quiz is

better than

that

of first quiz,

hence

it (^) is concluded that

he did^ better

in

second quiz.

Central

Limit

Theorenm Formula

The Central Limit

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 will

be

the population^

mean.

Similarly,

if you

find

the

average

of all^ of the

standard deviations^

in

your sample,

you (^) will

find

the actualstandard deviation^

for

your

population.

Mean of sample is^

same (^) as

the

mean

of the

population.

The

standard deviation^

of the sample is

equal to

the standard deviation^

of

the

population divided by the

square root of

the

sample

size.

Central limit^ theorem

is

applicable for

a

sufficiently large sample sizes (n 30).^ The

formula for^ central limit^ theorem

can

be

stated as^ follows:

and

Where,

=Population

mean

Vn

G= Population^

standard deviation

=

Sample

mean

=

Sample

standard deviation

n=

Sample

size

Solved Example

Question: The^ record of^ weights^ of^ the male

population follows (^) the normal distribution.

Its

mean and

standard deviations^

are 70 kg and

15 kg

respectively.

If (^) a

researcher considers the records

of 50 males, then what would^ be the

mean

and

standard deviation^

of the chosen sample?

Solution:

Mean

of

the

population (^) =

kg

Standard deviation^

of the population^

= 15 kg

sample

size

n =^

50

Mean of the

sample is^ given by:

=

70 kg

Standard deviation^

of the sample^

is

given by:

V

=

kg (approx)