Here, we list all Statistics Formulas for your use. 

 

The formula list includes

 

Range

Range = Maximum value – Minimum Value

 

For data

5, 10, 15, 18, 21, 24, 24, 13, 2, 9, 9, 18, 18, 32, 28, 3, 14, 25, 3, 27

 

Maximum value = 32

Minimum value = 2

 

Thus,

Range = 32 – 2 = 30

 

Mean of Raw, Discrete and Grouped Data

Mean of Raw, Discrete and Grouped Data.jpg

Median of Raw, Discrete and Grouped Data

Median of Raw, Discrete and Grouped Data.jpg

We can also find median using Ogive curve

We draw less than and more than ogive,

 and their intersection is the median

Mode of Raw, Discrete and Grouped Data

Mode of Raw, Discrete and Grouped Data.jpg

Mean deviation about Mean and Median

Mean deviation about Mean and Median.jpg

 

Variance and Standard Deviation

Variance and Standard Deviation.jpg

 

Coefficient of Variation

 

Coefficient of Variation.jpg

  1. Chapter 15 Class 11 Statistics
  2. Concept Wise

Transcript

Mean of Raw Data π‘₯ Μ… = (π‘†π‘’π‘š π‘œπ‘“ π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘ )/(π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›π‘ ) Mean of Discrete Data π‘₯ Μ… = (βˆ‘β–’γ€–π‘“_𝑖 π‘₯_𝑖 γ€—)/(βˆ‘β–’π‘“_𝑖 ) Mean of Grouped Data Direct Method π‘₯_𝑖 = (π‘ˆπ‘π‘π‘’π‘Ÿ π‘π‘™π‘Žπ‘ π‘  π‘™π‘–π‘šπ‘–π‘‘ + πΏπ‘œπ‘€π‘’π‘Ÿ π‘π‘™π‘Žπ‘ π‘  π‘™π‘–π‘šπ‘–π‘‘)/2 Mean = π‘₯ Μ… = (βˆ‘β–’γ€–π‘“_𝑖 π‘₯_𝑖 γ€—)/(βˆ‘β–’π‘“_𝑖 ) Assumed Mean Method a β†’ middle value of π‘₯_𝑖 column 𝑑_𝑖 = π‘₯_𝑖 βˆ’ a Mean = π‘₯ Μ… = a + (βˆ‘β–’γ€–π‘“_𝑖 𝑑_𝑖 γ€—)/(βˆ‘β–’π‘“_𝑖 ) Step-Deviation Method 𝑒_𝑖 = (π‘₯_𝑖 βˆ’ π‘Ž)/β„Ž h β†’ class size Mean = π‘₯ Μ… = a + (βˆ‘β–’γ€–π‘“_𝑖 𝑒_𝑖 γ€—)/(βˆ‘β–’π‘“_𝑖 ) Γ— h Median Median of Raw Data Write data in ascending/descending order N = Total Number of observations (i) If N is odd Median = ((𝑁 + 1)/2)^π‘‘β„Ž observation (ii) If N is even Median = ((𝑁/2)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘› + (𝑁/2 + 1)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›)/2 Median of Discrete Data First, we find cumulative frequency (cf) Then, we find 𝑁/2 N = Total number of observations (i) If N is odd Median = Observation where cf is greater than 𝑁/2 (ii) If N is even (𝑁/2)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘› = Observation where cf is equal to 𝑁/2 (𝑁/2 + 1)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘› = Observation where cf is greater than 𝑁/2 Median = ((𝑁/2)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘› + (𝑁/2 + 1)^π‘‘β„Ž π‘œπ‘π‘ π‘’π‘Ÿπ‘£π‘Žπ‘‘π‘–π‘œπ‘›)/2 Median of Grouped Data We use the formula Median = l + ((𝑛/2 βˆ’ 𝑐𝑓))/𝑓 Γ— h where Median class = class with cumulative frequency greater than 𝑁/2 l = lower limit of median class h = class size f = frequency of median class cf = cumulative frequency of class preceding median class Mode Mode of Raw Data We follow these steps Arrange in ascending or descending order Find the element occurring max number of times ∴ Mode = Element occurring maximum number of times Mode of Discrete Data Mode is the data which occurs maximum number of times, ∴ Mode = Data with maximum frequency (fi) Mode of Grouped data We use the formula Mode = l + ((𝑓_1βˆ’ 𝑓_0)/(2 𝑓_1 βˆ’ 𝑓_(0 )βˆ’ 𝑓_2 )) Γ— h where Modal Class = Class with highest frequency l = lower limit of modal class f = frequency of modal class 𝑓_0 = frequency of class preceding modal class 𝑓_2 = frequency of class succeeding modal class h = class size Mean deviation of Raw Data Raw data is like π‘₯_1, π‘₯_2, π‘₯_3, … … π‘₯_𝑛 Mean Deviation About Mean = 1/𝑛 βˆ‘_(𝑖=1)^𝑛▒〖|π‘₯_𝑖 βˆ’ π‘₯ Μ…|γ€— where π‘₯ Μ… = mean Mean Deviation About Median = 1/𝑛 βˆ‘_(𝑖=1)^𝑛▒〖|π‘₯_𝑖 βˆ’π‘€|γ€— where M = Median Mean deviation of Discrete Data Discrete data looks like x x1 x2 x3 x4 x5 . . . xn f f1 f2 f3 f4 f5 . . . fn Mean Deviation about mean = 1/𝑁 βˆ‘_(𝑖 = 1)^𝑛▒〖𝑓_𝑖 |π‘₯_𝑖 βˆ’ π‘₯ Μ…|γ€— where N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 and π‘₯ Μ… = mean = (βˆ‘β–’γ€–π‘“_𝑖 π‘₯_𝑖 γ€—)/(βˆ‘β–’π‘“_𝑖 ) Mean Deviation about median = 1/𝑁 βˆ‘_(𝑖 = 1)^𝑛▒〖𝑓_𝑖 |π‘₯_π‘–βˆ’π‘€|γ€— where N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 and M = Median Mean deviation of grouped continuous data Mean Deviation About Mean π‘₯_𝑖 = (π‘ˆπ‘π‘π‘’π‘Ÿ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ π‘π‘™π‘Žπ‘ π‘  + πΏπ‘œπ‘€π‘’π‘Ÿ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ π‘π‘™π‘Žπ‘ π‘ )/2 Mean deviation about mean = 1/𝑁 βˆ‘_(𝑖 = 1)^𝑛▒〖𝑓_𝑖 |π‘₯_𝑖 βˆ’ π‘₯ Μ…|γ€— where N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 and π‘₯ Μ… = mean We can find mean by normal method or step-deviation method Mean Deviation about median = 1/𝑁 βˆ‘_(𝑖 = 1)^𝑛▒〖𝑓_𝑖 |π‘₯_π‘–βˆ’π‘€|γ€— where N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 and M = Median = l + ((𝑁/2 βˆ’ 𝐢)/𝑓) Γ— h Variance and Standard Deviation for raw data Variance 𝜎^2 = 1/𝑛 βˆ‘_(𝑖 = 1)^𝑛▒(π‘₯_π‘–βˆ’π‘₯ Μ… )^2 where π‘₯ Μ… = mean Standard deviation 𝜎 = βˆšπ‘‰π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘π‘’ 𝜎 = √(1/𝑛 (βˆ‘β–’(π‘₯_π‘–βˆ’π‘₯ Μ… )^2 )) where π‘₯ Μ… = mean Variance and Standard deviation for Discrete Frequency Distribution Standard deviation (𝜎) = √(1/𝑁 βˆ‘β–’γ€–π‘“_𝑖 (π‘₯_π‘–βˆ’π‘₯ Μ… )γ€—^2 ) where π‘₯ Μ… = mean N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 Note:- Variance = (π‘†π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π·π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘›)^2 Variance and Standard deviation for Grouped Frequency Distribution We can use different methods to find Standard Deviation Normal method to find Standard deviation Standard deviation (𝜎) = √(1/𝑁 βˆ‘β–’γ€–π‘“_𝑖 (π‘₯_π‘–βˆ’π‘₯ Μ… )^2 γ€—) where π‘₯ Μ… = mean N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 Another method to find Standard Deviation 𝜎 = 1/𝑁 √(π‘βˆ‘β–’γ€–π‘“_𝑖 γ€–π‘₯_𝑖〗^2 γ€—βˆ’(βˆ‘β–’γ€–π‘“_𝑖 π‘₯_𝑖 γ€—)^2 ) where N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 Shortcut method to find Standard Deviation 𝜎 = β„Ž/𝑁 √(π‘βˆ‘β–’γ€–π‘“_𝑖 〖𝑦_𝑖〗^2 γ€—βˆ’(βˆ‘β–’γ€–π‘“_𝑖 𝑦_𝑖 γ€—)^2 ) where 𝑦_𝑖 = (π‘₯_𝑖 βˆ’ π‘Ž)/β„Ž a = Assumed mean (the middle most value of xi) N = βˆ‘_(𝑖 = 1)^𝑛▒𝑓_𝑖 Coefficient of Variation Coefficient of variation (C.V) = 𝜎/π‘₯ Μ… Γ— 100 where 𝜎 β†’ standard deviation π‘₯ Μ… β†’ mean Less C.V means more consistent data

About the Author

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.