Measure of Central Tendency Flashcards
Arithmetic Mean
i] Discrete
ii] Discrete with frequency
iii] Grouped Discrete data with frequency.
I] E xi /n
ii] E xifi / Efi
iii] A + E df /Ef x h ; A is assumed mean and H is class length. d= X - A / h
Sum of Deviation of a set of observations from their AM is
E(xi - /x) = 0
Efi(xi -/x) = 0
AM is affected by change of origin and Change of scale then Formula
y = a +bx
Combined AM
n1x1 + n2x2 /n1 + n2
Median
Discrete
1. ODD
2. EVEN
- (n+1/2 )th Observation
- (n/2) th observation + (n/2+1) th observation / 2
Median Continuous
l + N/2 - CF / f x c ; l - lower limit , Cf - preceding CF , h- Class interval
Discontinuous [ add or sub amount]
half of the discontinuous amount
Median if x and y are two variables
y = a+bx
Deviations in Median
E|x-A| is minimum
Q,D,P
___ observation + 0.___ x Difference between the ___ and ___ values .
Q= 1/ 4
d = 1/10
p = 1/100
mode
l [ f0 -f-1/ 2f0 - f-1 - f1] x C ;
l LCB model class ;
f0 - frequency of the modal class
f1 - Frequency of post modal class
f-1 - Frequency of pre modal class
C - class length of the Modal class
Mode with respect to median and Mean
Mean - Mode = 3(mean - Median)
GM formula
( x1 * x2x3x4…xn) 1/n
GM properties
- if all the properties of GM are k
- GM of z =
- k
- GM of x * GM of y
Harmonic mean formula and Properties
n / E(1/x)
Grouped data
N/E[f/x]
If all the observations are taken as K, HM is K
n1+n2/ n1/h1 + n2/h2
Mode is affected by ?
Sampling fluctuations and not by mathematical property
Dispersion
Meaning ?
Amount of deviation from central frequency
Measure of Dispersion
Absolute measures
Range
MD
QD
SD
Measure of Dispersion
Relative Measures
Coefficient of Range
Coefficient of MD
Coefficient of QD
Coefficient of SD
Range
Meaning
formulas
For every grouped / ungrouped data
Range = Largest - Smallest
Coefficient of Range = L - S / L+ S x 100
Mean Deviation
Ungrouped data
1. Mean
2. Median
3. Assumed Mean
- E|x-x’|/n
- E|x-M|/n
- E|x-A|/n
Mean Deviation
Grouped Data [Both discrete and continuous]
- Mean
- Median
- Assumed Mean
and coefficient of MD
- Ef(x-x’) / n
- Ef(x-M) / n
- Ef(x-A) / n
Coefficient of MD
1. mean - MD about X’/x’ * 100
2. Median - MD about A / A * 100
3. Assumed Mean - MD about M / M * 100
Standard Deviation
Meaning
For Ungrouped Data
For Grouped Data
It takes AM , Best Measure of Dispersion
Un Grouped Data
SD = Square root (Ex^2/n - (X’)^2) , X’ = Ex/n
or
SD = Square root (E(x-x’)^2 / n)
Grouped data
SD = Square root (Efx^2/N - (X’) ^2)
Properties of SD
SD of any 2 no’s A and B
SD of a and b = |a-b/2| = Range /2
SD remain unaltered on Shift of origin
-> If You add/Subract a number with all the x nothing will change
SD remain altered on Shift of Scale
-> Multiplying/ Dividing same number with all the no of X things
SD of First “n” natural Numbers
Square Root (n^2 -1 /12)
Variance =
Variance = (SD) ^2
[Square root [ E (x-x’)^2/n] = E(x-x’)^2 /n
Coefficient of variation [CV]
a. Lesser CV _______ consistency
b. Greater CV ________ Consistency
Where CV used?
SD / AM * 100
a. Greater Consistency
b. Worst Consistency
CV is a statistical tool to test the consistency of the data
combined SD
Square Root ( n1s1^2 + n2 S2^2 + n1d1 ^2 + n2d2 ^2 / n1+n2)
d1 ^2 = (X’1 - X’c) ^2
d2^2 = (X’2 - X’c) ^2 ,,
X’c = Combined Mean
X’c = n1x1’ + n2x2’ /n1+n2
QD - Quartile Deviation
Meaning
QD - formula
Coefficient of QD
Appropriate measure of dispersion for open end classification
QD = Q3 - Q1 / 2
Q3 -Q1 = Inter Quartile Range
Q3-Q1/2 = Semi Inter Quartile Range
Coefficient of QD = Q3 - Q1 /Q3+Q1 x 100
or QD / M * 100 , M -> Median
Relationship between SD , MD , QD
SD > MD > QD
Example
2SD = 2.5MD = 3QD
if it’s all are equated so one having highest multiplier is smallest and One having lowest Multiplier is biggest
x and y are 2 variable as ax + by + c =0
with dispersion of x then
dispersion of y = |a/b| * Dispersion of X
Best Measure of central tendency without any conditions
AM
Best measure
Positional average
median
For Open end classification best measure
Median and partition
Which means of central tendency are suitable for % and Ratios
GM
Which are ratios
HM and GM
