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Formula Sheet for Statistics, Cheat Sheet of Statistics

Formula Sheet for ST 311 at NCSU

Typology: Cheat Sheet

2024/2025

Uploaded on 01/05/2025

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Exploring Data: Distributions
โ€ข
Evaluate overall pattern (shape, center, spread) and deviations
(outliers).
โ€ข
Mean:
๐‘ฅ๎ชง=โˆ‘๐‘ฅ
๐‘›=๐‘ฅ1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›
๐‘›
โ€ข
Standard deviation:
๐‘ =โˆšโˆ‘(๐‘ฅโˆ’๐‘ฅ๎ชง)2
๐‘›โˆ’1
โ€ข
Quartiles:
o
The first quartile Q1 is the point with 25% of the data below it.
o
The median M is the point with 50% of the data below it.
o
The third quartile Q3 is the point with 25% of the data above it.
โ€ข
Interquartile range: IQR = Q3 - Q1
โ€ข
Five-number summary:
o
Minimum, Q1, M, Q3, Maximum
โ€ข
If X has mean ยต and standard deviation ฯƒ:
๐‘ง=๐‘ฅโˆ’๐œ‡
๐œŽ and ๐‘ฅ = ๐œ‡ + ๐‘งโˆ— ๐œŽ
Binomial Distribution:
โ€ข
๐ธ(๐‘‹)=๐‘›๐‘
โ€ข
๐‘†๐ท(๐‘‹)=โˆš๐‘›๐‘๐‘ž
Sampling Distribution of a Sample Proportion
โ€ข
The sample proportion, ๐‘๎žธ, is
๐‘๎žธ=๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ ๐‘ข๐‘๐‘๐‘’๐‘ ๐‘ ๐‘’๐‘ 
๐‘ ๐‘Ž๐‘š๐‘๐‘™๐‘’ ๐‘ ๐‘–๐‘ง๐‘’ =๐‘ฅ
๐‘›
โ€ข
๐œ‡๐‘๏œ=๐ธ(๐‘๎žธ)=๐‘
โ€ข
๐œŽ๐‘๏œ=๐‘†๐ท(๐‘๎žธ)=โˆš๐‘๐‘ž
๐‘›
โ€ข
The distribution of ๐‘๎žธ can be approximated by a Normal if
โ–ช The sample is random, and
โ–ช ๐‘›๐‘โ‰ฅ10 ๐’‚๐’๐’… ๐‘›๐‘žโ‰ฅ10
โ€ข
The standardized value of ๐‘๎žธ is ๐‘ง=๐‘๏œโˆ’๐‘
โˆš๐‘๐‘ž
๐‘›
Sampling Distribution of a Sample Mean
โ€ข
๐œ‡๐‘‹
๏Œค=๐ธ(๐‘‹
๏Œค)=๐œ‡
โ€ข
๐œŽ๐‘‹
๏Œค=๐‘†๐ท(๐‘‹
๏Œค)=๐œŽ
โˆš๐‘›
โ€ข
The distribution of ๐‘‹
๏Œค can be approximated by a Normal if
โ–ช The sample is random, and
โ–ช The CLT applies:
โ€ข Parent population is Normal, or
โ€ข ๐‘›โ‰ฅ30
โ€ข
The standardized value of ๐‘‹
๏Œค is ๐‘ง=๐‘ฅ๎ชงโˆ’๐œ‡
๐œŽ
โˆš๐‘›
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Exploring Data: Distributions

  • Evaluate overall pattern (shape, center, spread) and deviations

(outliers).

  • Mean:

1

2

๐‘›

  • Standard deviation:

2

  • Quartiles:

o The first quartile Q1 is the point with 25% of the data below it.

o The median M is the point with 50% of the data below it.

o The third quartile Q3 is the point with 25% of the data above it.

  • Interquartile range: IQR = Q 3

- Q

1

  • Five-number summary:

o Minimum, Q 1

, M , Q

3

, Maximum

  • If X has mean ฮผ and standard deviation ฯƒ:

๐‘ฅโˆ’๐œ‡

๐œŽ

and ๐‘ฅ = ๐œ‡ + ๐‘ง โˆ— ๐œŽ

Binomial Distribution:

Sampling Distribution of a Sample Proportion

  • The sample proportion, ๐‘ฬ‚ , is

๐‘ฬ‚

๐‘ฬ‚

๐‘๐‘ž

๐‘›

  • The distribution of ๐‘ฬ‚ can be approximated by a Normal if

โ–ช The sample is random, and

  • The standardized value of ๐‘ฬ‚ is ๐‘ง =

๐‘

ฬ‚ โˆ’๐‘

โˆš

๐‘๐‘ž

๐‘›

Sampling Distribution of a Sample Mean

๐‘‹

ฬ…

๐‘‹

ฬ…

๐œŽ

โˆš๐‘›

  • The distribution of ๐‘‹

can be approximated by a Normal if

โ–ช The sample is random, and

โ–ช The CLT applies:

  • Parent population is Normal, or
  • The standardized value of ๐‘‹

is ๐‘ง =

๐‘ฅฬ… โˆ’๐œ‡

๐œŽ

โˆš๐‘›

Confidence Intervals for a Proportion

  • The conditions to perform a CI for ๐‘:

โ–ช The sample is random, and

  • CI formula:

๐›ผ

2

Hypothesis Test for a Proportion

  • The conditions to perform a HT for ๐‘:

โ–ช The sample is random, and

  • Test statistic formula:

โˆ—

Confidence Intervals for a Mean

  • The conditions to perform a CI for ๐œ‡:

โ–ช The sample is random, and

โ–ช The CLT applies:

  • Parent population is Normal, or
  • CI formula: ๐‘ฅฬ… ยฑ |๐‘ก

๐›ผ

2

,๐‘‘๐‘“

๐‘ 

โˆš

๐‘›

Hypothesis Test for a Mean

  • The conditions to perform a HT for ๐œ‡:

โ–ช The sample is random, and

โ–ช The CLT applies:

  • Parent population is Normal, or
  • Test statistic formula:

โˆ—

Hypothesis Test for a Mean Difference

  • The conditions to perform a HT for ๐œ‡

๐‘‘

โ–ช The samples are random,

โ–ช The samples are dependent, and

โ–ช The CLT applies:

  • Parent population is Normal, or
  • Test statistic formula:

โˆ—

๐‘‘

๐‘‘

Common Values from the Standard Normal

  1. 05
  1. 025
  1. 01
  1. 005