Basic stats information

7/30/2019 Basic stats information

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Main topic Topic What it says

Histogram

summarize data and present in graphical

form. Shows the number of times the

data points repeat on various frequency

OutliersUnderstand what it implies - Decide toleave it or remove from the dataset

Mean

shows the average value of the data set.

Outliers can distort the mean value

easily and skew the data towards it.

Weighs the value of every data

when plotting histogram if the mean is

moved towards the right of the graph it

means that outliers are pulling the mean

towards it

Median

middle value of the data set when

arranged in ascending order.

Answer to problem when the data is

skewed due to outliers

Mode

Most frequently ocurring value in data

set. Use this when value is not important

When data has two modes, its called

bimodal distribution

Standard deviation

measures the level of

dispersion(variability) in the given set ofdata

Std dev tells us how far the data set

represent from mean. Higher std dev

means data is widely spread from mean.

Lower std dev means data closer to

mean

Coeffcient of variation mesured at std dev divided by mean

Two variables

Scatter plot - visual summary of

relationship between variables

when one variable is time, relation ship

is called time series

false relationship could be purely due to

co incidences. Look out for hidden

variables

scatter plot - does not prove casuality,

never prove one variable cuase the other


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Factors that affect interval level

- Sample mean should be at center of the

range

- higher std dev greater uncertainity

about population, wider range to bring in

confidence- small sample size demand wider range

to create confidence that pm is within

the SM

- More confident we want our SM

represent PM, wider would be the range

Normal

Distribution

- Shape of bell curve with mean at the

center

- X axis is the variable we are studying

and Y axis is the likelyhood of different

value that occurs

whats special

mean and median are the same.

Probabiility of value less than mean is

50% and more than mean is 50%

- location, widthness and narrowness of

the curve depends on the std deviation

and mean

importance of std dev

Large std dev makes the curve flat, small

std dev makes the curve narrow and tall

(with values more close to mean)

rule of thumb

68% of the time, the range lies within 1std dev from the mean

95% of probability, range lies within 2 std

dev from the mean

Z value

translates any value in to corresponding

Z value by subracting the mean and

divide by std dev

z multipliedby std dev and add/subract

from mean would give range and the

probability within which range is present

(68%, 95%, 99%)

if we start from very left of the curve

then it measures cumulative probability.

Probability works only on normal

distribution curve (not on all the curves)

a


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How to find cummulative probability

first standardize the value of the variable

by using excel standardize function (this

will find out the value of Z). Second use

norms dist function to find out the

cummulative probability

Other option is to use normdist withvalue of True. This wil return the

cummulative probability

How to find Z value if you have

cummulative prob value

How to find value of the variable if you

have cummulative prob, sample mean

and std dev

Central Limit Theorom

sample mean distributed approximately

normally regardless of distribution of the

population

more samples, better approximation of

normal distribution

Mean distribution of sample =

population

Properties of normal distribution to

extract info from sample

Confidence

intervals Estimating population mean

It's important to emphasize: We are not

saying that 95% of the time our sample

mean is the population mean, but we aresaying that 95% of the time a range that

is two standard deviations wide centered

around the sample mean contains the

population mean.

Increase confidence level

accept higher range or increase sample

size

How wide the interval

How do we know if an interval is too

wide? Typically, if we would make a

different decision for different values

within an interval, that interval is too

wide.

std dev of sample mean

How to find confidence interval this works only if the sample size is > 30

need to know the level of confidence


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Obtaining Z value

Converting the desired confidence level

into the corresponding cumulative

probability on the standard normal curve

is essential because Excel's NORMSINV

function and the z-table work with

cumulative probabilitiesFor smaller sample size (less than 30) we

have to use T value Degree of freedom = sample size -1

Choosing sample size

based on initial estimate, find out sd,

also find out what should be the

maximum deviation allowed. Apply

following formula to get the desired

sample size

summary of how to build the range that

constitutes population mean

working with proportions

often used to indicate frequency of some

phenomenon in the population

p bar is the proportion of yes to a total

population

sample size selection

selected sample size should satisfy the

condition mentioned


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Method Calculations/Formulas

Use excel function (under analysis tool

pak)

Greek letter mu represent mean of data

aset user average formula in excel

use median formula in excel

use mode formula in excel

Greem letter sigma Use excel formula STDEV

can be used to compare among different

set of data


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use excel correl function to find out the

correlation

- select elements from population at

random

- Analyze the sample

- Draw inference about total population

we are interested in

Need to know x bar (sample mean), std

dev of sampel s and sample siize n. Z

represent confidence level. Higher value

of Z higher the confidence level is


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standardize, normsdist

Normdist

Normsinv 2.807033768

Norminv

std dev of population mean divided by sq

root of n


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to convert desired conf level, take 1-

desired conf level and divide by 2. Then

add the result to the desire conf level.Input 1-confidence interval and degree

of freedom Use TINV

solve the equation or use the excel utility

use the excel utility - confidence interval

use excel utility

n x p bar >= 5, n x (1- p bar) > = 5

number of rooms available divided by

upper limit of the confidence leve


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Type of Estimate:

Sample Size:

Input Area n 70 n 20 n 100x-bar 4.5 x-bar 5 p-bar 0.1s 1.2 s 10 confidence level 0.95confidence level 0.95 confidence level 0.95

Output Area Center of Interval 4.50 Center of Interval 5.00 Center of Interval 0.10z*s/sqrt(n) 0.28 t*s/sqrt(n) 4.68 z*s/sqrt(n) 0.06

Lower end of int'l 4.22 Lower end of int'l 0.32 Lower end of int'l 0.041

Upper end of int'l 4.78 Upper end of int'l 9.68 Upper end of int'l 0.159

Other Interval width 0.56 Interval width 9.36 Interval width 0.12

Calculations 1-confidence level 0.05 1-conf 0.05 1-confidence level 0.05(1-confidence level)/2 0.025 t 2.09 (1-confidence level)/2 0.025z 1.96 z 1.96

sqrt(n) 8.37 sqrt(n) 4.47 (p)(1-p) 0.09s/sqrt(n) 0.14 s/sqrt(n) 2.24 s = sqrt[(p)(1-p)} 0.30

sqrt(n) 10.00s/sqrt(n) 0.03

Check assumptions:np>5 OKn(1-p) > 5 OK

Confidence Interval Utility

Mean Proportions

n >= 30 n < 30 n >= 30

146230062.xlsx.ms_office Confidence Intervals


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Type of Estimate:

Input Area Sample Standard Deviation, s 50 Estimate of p 0.1Desired Accuracy: Half Width of Interval, d 5 Desired Accuracy: Half Width of Interval, d 0.02Confidence level 0.95 Confidence level 0.95

Output Area Required Sample Size 385 Required Sample Size 865

Other 1-confidence level 0.05 1-confidence level 0.05Calculations (1-confidence level)/2 0.025 (1-confidence level)/2 0.025

z 1.96 z 1.96

z*s 98.00 (p)(1-p) 0.09z*s/d 19.60 s = sqrt[(p)(1-p)} 0.30Minimal n 384.15 z*s = {z*sqrt[(p)(1-p)]} 0.59

z*s/d = {z*sqrt[(p)(1-p)]}/d 29.40Minimal n to ensure np>5 50.0Minimal n to ensure n(1-p)>5 5.6Minimal n to ensure d < (zs/sqrt(n)) 864.3Minimal n to satisfy all constraints 864.3

Assumptions:

Sample Size will be above 30. If not, raise sample size to 30 to make assumptions valid.Proportion Estimate is the maximum you expect p to be. If you don't have a good estimate of the proportion,use p = .5, which gives maximal standard deviation.

Sample Size Utility

Mean Proportion

146230062.xlsx.ms_office Sample Size


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Cereal

Protein

(grams

per

serving)

Carbohydr

ates

(grams

per

serving)

100% Bran 1 12

All-Bran 1 12Almond Delight 1 12

Apple Cinnamon Cheerios 1 13

Apple Jacks 1 13

Bran Chex 1 13

Bran Flakes 1 13

Cap'n'Crunch 1 13

Cheerios 1 14

Cinnamon Toast Crunch 1 14

Cocoa Puffs 1 15

Corn Chex 1 23

Corn Flakes 2 9

Corn Pops 2 10

Count Chocula 2 11

Cracklin' Oat Bran 2 11

Cream of Wheat (Quick) 2 11

Crispix 2 12

Double Chex 2 14

Froot Loops 2 15

Frosted Flakes 2 15

Frosted Mini-Wheats 2 15

Fruit & Fibre Dates, Walnuts, and Oats 2 16Fruity Pebbles 2 18

Golden Grahams 2 21

Grape Nuts Flakes 2 21

Grape-Nuts 2 21

Great Grains Pecan 2 21

Honey Nut Cheerios 2 21

Honey-comb 2 22

Kix 2 22

Life 3 10

Lucky Charms 3 11

Maypo 3 11

Muesli Raisins, Dates, & Almonds 3 12

Muesli Raisins, Peaches, & Pecans 3 12

Mueslix Crispy Blend 3 13

Nut&Honey Crunch 3 13

Nutri-grain Wheat 3 14

Post Nat. Raisin Bran 3 14

Product 19 3 15


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Puffed Rice 3 15

Puffed Wheat 3 16

Quaker Oat Squares 3 17

Raisin Bran 3 17

Raisin Nut Bran 3 17

Raisin Squares 3 17

Rice Chex 3 18Rice Krispies 3 20

Shredded Wheat 3 21

Smacks 4 5

Special K 4 7

Total Corn Flakes 4 12

Total Raisin Bran 4 14

Total Whole Grain 4 16

Triples 4 16

Trix 4 16

Wheat Chex 6 16

Wheaties 6 17

mean 2.49 14.81

median 2.00 14.00


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Variable 1 Variable 2 Age

a ary($thousan

ds)

-1.0 1.0 53 145

-1.0 1.0 43 621

1.0 -1.0 33 262

1.0 -1.0 45 208

-1.0 1.0 46 362

1.0 -1.0 55 424

-1.0 1.0 41 339

-1.0 1.0 55 736

-1.0 1.0 36 291

1.0 -1.0 45 58

1.0 -1.0 55 498

1.0 -1.0 50 643

49 390

-1.000000 47 332

69 75051 368

48 659

62 234

45 396

37 300

50 343

50 536

50 543

58 217

53 298

57 1103

53 406

61 254

47 862

56 204

44 206

46 250

58 21

48 298

38 350

74 80060 726

32 370

51 536

50 291

40 808

61 543

63 149

56 350


16/16

45 242

61 198

70 213

59 296

57 317

69 482

44 15556 802

50 200

56 282

43 573

48 388

52 250

62 396

48 572

0.13

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Basic stats information