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# Statistics formulae

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• 1. STATISTICS FORMULAEMEASURES OF CENTRAL TENDENCY: 1) Range = Highest Value &#x2013; Lowest Value 2) No. Of Class Intervals = 1+ 3.2 log N (Struge&#x2019;s Rule) 3) Class Width = Range / No. Of Class Intervals. OR Class Intervals = Range / Class Width 4) Arithmetic Mean = &#x2211; &#x202B;/ &#x754;&#x202C;n, OR &#x2211;&#x742;&#x202B; / &#x754;&#x202C;N &#xC4A; &#xB3F; &#xB54;.&#xB47;.&#xB4A; L+&#x1248; &#x1249;&#xD7;H &#xB6C;&#xB3E;&#xB35; &#xC2E; &#xB36; &#xB4A; 5) Median = or Where: L = Lower limit of median class, PCF = Previous Cumulative Frequency, F= Frequency of the Median Class, &amp; H= Width of the CI &#x202B;+&#x72E;&#x202C;&#x1242; &#x1243;&#xD7;&#x210E; &#xBD9;&#xB35;&#xB3F;&#xBD9;&#xB34; &#xB36;&#x123A;&#xBD9;&#xB35;&#x123B;&#xB3F;&#xBD9;&#xBE2;&#xB3F;&#xBD9;&#xB36; 6) Mode = Where: Where: L = Lower limit of modal class, f1 = Highest Frequency in the Distribution, f0 = Preceding Frequency to the Highest Frequency in the Distribution, f2 = Succeeding Frequency to the Highest Frequency in the Distribution, &amp; H= Width of the CI &#xBE1; &#xB52; &#xC2D; &#xC5C; &#x2211; &#x2211; 7) Harmonic Mean = or &#xCE3; &#xC6E; &#x2211; &#xBC5;&#xBE2;&#xBDA; &#xBEB; &#x2211; &#xBD9;&#xBC5;&#xBE2;&#xBDA; &#xBEB; 8) Geometric Mean = &#x202B;&#x743;&#x74B;&#x748;&#x745;&#x750;&#x74A;&#x723;&#x202C; &#x202B;&#x743;&#x74B;&#x748;&#x745;&#x750;&#x74A;&#x723;&#x202C; &#xBE1; &#xBC7; orPraveenKumar Keskar 1
• 2. &#xCBF; &#xBE5;&#x1240; &#xC30; &#x1241;&#xB3F;&#xBC9;.&#xBBC;.&#xBBF; &#x202B;&#x74E;&#x202C;&#x1240; &#x1241; &#x202B;&#x74E;&#x202C;&#x1240; &#x1241; &#x748;+&#x1248; &#x1249;&#xD7;&#x210E; &#xBE1;&#xB3E;&#xB35; &#xBC7;&#xB3E;&#xB35; &#xB38; &#xB38; &#xBBF; 9) Quartiles = or or &#x74A;+1 &#x730;+1 &#x202B;&#x74E;&#x202C;&#x1240; &#x1241; or &#x202B;&#x74E;&#x202C;&#x1240; &#x1241; 10 10 10) Deciles = or &#xCBF; &#xBE5;&#x1240; &#x1241;&#xB3F;&#xBC9;.&#xBBC;.&#xBBF; &#x748;+&#x1248; &#xC2D;&#xC2C; &#x1249;&#xD7;&#x210E; &#xBBF; &#x202B; &#x74E;&#x202C;&#x1240; &#xB35;&#xB34;&#xB34; &#x1241; &#x202B; &#x74E;&#x202C;&#x1240; &#xB35;&#xB34;&#xB34; &#x1241; &#xBE1;&#xB3E;&#xB35; &#xBC7;&#xB3E;&#xB35; 11) Percentile = or or &#xCBF; &#xBE5;&#x1240; &#x1241;&#xB3F;&#xBC9;.&#xBBC;.&#xBBF; &#x748;+&#x1248; &#xC2D;&#xC2C;&#xC2C; &#xBBF; &#x1249;&#xD7;&#x210E;MEASURES OF DISPERSION: 12) Range = Highest Value &#x2013; Lowest Value &#xB4C;&#xB67;&#xB65;&#xB66;&#xB63;&#xB71;&#xB72; &#xB5A;&#xB5F;&#xB6A;&#xB73;&#xB63; &#x2013; &#xB50;&#xB6D;&#xB75;&#xB63;&#xB71;&#xB72; &#xB5A;&#xB5F;&#xB6A;&#xB73;&#xB63; &#xB4C;&#xB67;&#xB65;&#xB66;&#xB63;&#xB71;&#xB72; &#xB5A;&#xB5F;&#xB6A;&#xB73;&#xB63;&#xB3E; &#xB50;&#xB6D;&#xB75;&#xB63;&#xB71;&#xB72; &#xB5A;&#xB5F;&#xB6A;&#xB73;&#xB63; Co-efficient of Range = &#xB55;&#xB37;&#xB3F;&#xBCA;&#xB35; &#xB36; 13) Quartile Deviation = &#xB55;&#xB37;&#xB3F;&#xBCA;&#xB35; &#xBCA;&#xB37;&#xB3E;&#xBCA;&#xB35; Co-efficient of Q.D = &#xD25; &#x2211; |&#xBEB;&#xB3F;&#xBEB;| &#xD25; &#x2211; &#xBD9;|&#xBEB;&#xB3F;&#xBEB;| &#xBE1; &#xBC7; 14) Mean Deviation = or &#xBC6;&#xBD8;&#xBD4;&#xBE1; &#xBBD;&#xBD8;&#xBE9;&#xBDC;&#xBD4;&#xBE7;&#xBDC;&#xBE2;&#xBE1; &#xBD9;&#xBE5;&#xBE2;&#xBE0; &#xBE0;&#xBD8;&#xBD4;&#xBE1; &#xBC6;&#xBD8;&#xBD4;&#xBE1; Co-efficient of MD =PraveenKumar Keskar 2
• 3. Standard Deviation = &#xDA7; &#xDA7; &#x2211;&#x123A;&#xBEB;&#xB3F;&#xBEB;&#x305; &#x123B;&#xC2E; &#x2211; &#xBD9;&#x123A;&#xBEB;&#xB3F;&#xBEB;&#x305; &#x123B;&#xC2E; &#xBE1; &#xBC7; 15) or OR &#x2211;x &#xB36; &#x2211; fx &#xB36; &#xDA7; &#x2211; x2 &#x2212;&#x1240; n&#x1241; &#xDA7; &#x2211; fx2 &#x2212;&#x1240; &#x1241; n &#xB52; &#xB52; or &#xD7; 100 &#xBA2; &#xD24; &#xB76; Co-efficient of Variation =PROBABILITY: 16) Additional Theorem = P&#x123A;A &#x22C3; B&#x123B; = P&#x123A;A&#x123B; + P&#x123A;B&#x123B; &#x2212; P&#x123A;A &#x22C2; B&#x123B; P &#x1246;B&#xD57;A&#x1247; = &#xB54;&#x123A;&#xB45; &#x22C2; &#xB46;&#x123B; &#xB54;&#x123A;&#xB45;&#x123B; 17) Conditional Theorem = &#xB54;&#x123A;&#xB45;&#x123B;&#xD7;&#xB54;&#x123A;&#xB45; Baye&#x2019;s Theorem = P &#xD6C;A&#xD57; &#xD70; = &#xD57;&#xB49; &#x123B; E&#xB35; &#xC2D; 18) &#xB45;&#xD57; &#x123B; &#x2211; &#xB54;&#x123A;&#xB49;&#xC2D; &#x123B;&#xD7;&#xB54;&#x123A; &#xB49; &#xC2D; 19) Binomial Distribution: Probability Mass Function (P.M.F) = n&#xB47;&#xC6E; &#xD7; &#x123A;p&#x123B;&#xB76; &#xD7; &#x123A;q&#x123B;&#xB6C;&#xB3F;&#xB76; Where, &#x202B;&#x754;&#x202C; = 0, 1, 2 _ _ _ n, p+q = 1 20) Poison Distribution &#x123A;&#xBD8; &#xC37;&#xCD8; &#x123B;&#xD7;&#x123A;&#xBE0;&#x123B;&#xCE3; &#xBEB;! P.M.F = Where, x = 0, 1, 2, 3 _ _ _ _ _ _ _ _ &#x221E;PraveenKumar Keskar 3
• 4. &#xBEB;&#xB3F;&#xBE0; &#xC19; 21) Normal Distribution = Where m = mean &amp; &#x7EA; = Standard DeviationCORRELATION: 22) Karl Pearson&#x2019;s Coefficient of Correlation: &#xBE1;&#xB8A;&#xBEB;&#xBEC;&#xB3F;&#x123A;&#xB8A;&#xBEB;&#x123B;&#x123A;&#xB8A;&#xB77;&#x123B; &#xDA5;&#xBE1;&#xB8A;&#xBEB; &#xC2E;&#xC37; &#x123A;&#xB8A;&#xBEB;&#x123B;&#xC2E; &#xDA5;&#xBE1;&#xB8A;&#xB77;&#xC2E;&#xC37; &#x123A;&#xB8A;&#xB77;&#x123B;&#xC2E; r= 23) Spearman&#x2019;s Rank Correlation: &#xB3A;&#x123A;&#xB8A;&#xB48;&#xC2E; &#x123B; r = 1- [&#xBE1;&#x123A;&#xBE1; &#xB3F;&#xB35;&#x123B;] &#xC2E; where D = R1 &#x2013; R2 If any rank is repeated then; &#xB3A;&#x123A;&#xB8A;&#xB48;&#xC2E; &#xB3E; &#xB47;.&#xB4A;.&#x123B; r=1&#x2013; [ &#xBE1;&#x123A;&#xBE1;&#xC2E; &#xB3F;&#xB35;&#x123B; ] &#xBE0;&#x123A;&#xBE0;&#xC2E; &#xB3F;&#xB35;&#x123B; &#xB35;&#xB36; C.F. = Where, m = No. of times the rank is repeatedREGRESSION: 24) If X is dependent on Y= &#xD24;) (&#x202B; = ) &#x305;&#x754;- &#x754;&#x202C;bxy (&#x202B;&#x755;- &#x755;&#x202C; &#xC19;&#xBEB; &#xC19;&#xBEC; bxy = r 25) If Y is dependent on X =PraveenKumar Keskar 4
• 5. &#xD24;) (y - &#x202B; = &#x755;&#x202C;byx (&#x202B;) &#x305;&#x754;- &#x754;&#x202C; &#xC19;&#xBEC; &#xC19;&#xBEB; byx = r &#xBE1;&#xB8A;&#xB76;&#xB77;&#xB3F;&#x123A;&#xB8A;&#xB76;&#x123B;&#x123A;&#xB8A;&#xB77;&#x123B; &#xBE1;&#x123A;&#xB8A;&#xB76;&#xC2E; &#x123B; &#xB3F; &#x123A;&#xB8A;&#xB76;&#x123B;&#xC2E; byx = &#xBE1;&#xB8A;&#xB76;&#xB77;&#xB3F;&#x123A;&#xB8A;&#xB76;&#x123B;&#x123A;&#xB8A;&#xB77;&#x123B; &#xBE1;&#x123A;&#xB8A;&#xB77;&#xC2E; &#x123B;&#xB3F; &#x123A;&#xB8A;&#xB77;&#x123B;&#xC2E; bxy = r = &#xDA5;&#x73E;&#xBEB;&#xBEC; &#xD7; &#x73E;&#xBEC;&#xBEB;TIME SERIES 26) Method of Moving Averages 27) Method of Least Squares a) Linear Equation: y = a + bx &#x2211;&#xBEC; &#x2211; &#xBEB;&#xBEC; &#xBE1; &#x2211; &#xBEB;&#xC2E; Where, a = ; b= b) Quadratic Equation: y = a + bx + cx2 Normal Equations are: &#x3A3;y = na + b&#x3A3;x + c&#x3A3;x2 &#x3A3;xy = a&#x3A3;x + b&#x3A3;x2 + c&#x3A3;x3 &#x3A3;x2y = a&#x3A3;x2 + b&#x3A3;x3 + c&#x3A3;x4INDEX NUMBERS 28) Unweighted Simple Average: a) Unweighted Arithmetic Mean Index Numbers = &#xB8A;&#xBC9; &#xBE1; P01 =PraveenKumar Keskar 5
• 6. b) Unweighted Geometric Mean Index Number = &#xB8A;&#xB6A;&#xB6D;&#xB65;&#xB54; &#xBE1; P01 = Antilog ( ) 29) Unweighted Simple Aggregative Index Numbers = &#xB8A;&#xB54;&#xC2D; &#xB8A;&#xB54;&#xC2C; P01 = &#xD7; 100 30) Weighted Average Relative Index Numbers: a) Weighted Arithmetic Mean Index Number = &#xB8A;&#xB5B;&#xB54; &#xB8A;&#xB5B; P01 = b) Weighted Geometric Mean Index Number = P01 = Antilog (&#xB8A;&#xB5B;&#xB6A;&#xB6D;&#xB65;&#xB54;) &#xB8A;&#xB5B; 31) Weighted Aggregative Index Number = a) Laspeyre&#x2019;s Price Index Number = &#xB8A;&#xB54;&#xC2D; &#xB55;&#xC2C; &#xB8A;&#xB54;&#xC2C; &#xB55;&#xC2C; P01 = &#xD7; 100 b) Paasche&#x2019;s Price Index Number = &#xB8A;&#xB54;&#xC2D; &#xB55;&#xC2D; &#xB8A;&#xB54;&#xC2C; &#xB55;&#xC2D; P01 = &#xD7; 100 c) Marshall Edgeworth&#x2019;s Price Index Number = &#xB8A;&#xB54;&#xC2D; &#xB55;&#xC2C; &#xB3E; &#xB8A;&#xB54;&#xC2D; &#xB55;&#xC2D; &#xB8A;&#xB54;&#xC2C; &#xB55;&#xC2C; &#xB3E; &#xB8A;&#xB54;&#xC2C; &#xB55;&#xC2D; P01 = &#xD7; 100 d) Fisher&#x2019;s Price Index Number = P01 = &#xDA5;P&#xB34;&#xB35; &#x123A;Laspeyre&#x2032;s&#x123B; &#xD7; P&#xB34;&#xB35; &#x123A;Paasche&#x2032;s&#x123B; = &#xDA8;&#x3A3;P1 Q0 &#xD7; &#x3A3;P1 Q1 &#x3A3;P Q &#x3A3;P Q 0 0 0 1PraveenKumar Keskar 6