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Descriptive Statistics - Simple Linear Regression - Example

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Free Statistics Calculator

Our calculator allows you to compute Simple Linear Regression statistics for any pair of data series.

Data

Example Data Set & Computation ()

Download

MS Excel Sheet.

Output

Observations Computation
# y x 1 2 3 4 5 6 7 8
1 3 1 -5 -2 25 4 10 0.0 0.00 0.00
2 5 2 -3 -1 9 1 3 -0.5 0.25 0.50
3 7 3 -1 0 1 0 0 -1.0 1.00 1.00
4 14 4 6 1 36 1 6 3.5 12.25 3.50
5 11 5 3 2 9 4 6 -2.0 4.00 2.00
Sum 40 15 0 0 80 10 25 0.0 17.50 7.00

Descriptive Statistics - Simple Linear Regression - Example

Computations 2

+----------------+
 COEFFICIENTS 1 
+----------------+

Variable        Coefficient     Stand.Err.        t-Ratio    Probability

INTERCEPT           .500000
St1X5              2.500000        .763763       3.273267        .023331
+----------------+
 COEFFICIENTS 2 
+----------------+

Variable         Elasticity     Stand.Err.        t-Ratio    Probability

St1X5               .937500        .286411       -.218218        .579370
+----------------+
 COEFFICIENTS 3 
+----------------+

Variable        Coefficient    Stand.Coef.     Elasticity     Rel.Contr.

INTERCEPT           .500000
St1X5              2.500000        .883883        .937500       1.000000
+----------------------------------+
 ANALYSIS OF VARIANCE 1 : GENERAL 
+----------------------------------+

ANOVA        df          Sum of Squares         Mean Square

Mean          1              320.000000
Regression    1               62.500000           62.500000
Residual      3               17.500000            5.833330

Total         5              400.000000           80.000000


F-Test Statistic      Value :     10.714300
       Degrees of Freedom 1 :      1
       Degrees of Freedom 2 :      3
                Probability :       .046662
+------------------------------------+
 ANALYSIS OF VARIANCE 2 : FROM MEAN 
+------------------------------------+

ANOVA        df          Sum of Squares         Mean Square

Regression    1               62.500000           62.500000
Residual      3               17.500000            5.833330

Total         4               80.000000           20.000000


F-Test Statistic      Value :     10.714300
       Degrees of Freedom 1 :      1
       Degrees of Freedom 2 :      3
                Probability :       .046662
+------------------------------------+
 ANALYSIS OF VARIANCE 3 : FROM ZERO 
+------------------------------------+

ANOVA        df          Sum of Squares         Mean Square

Regression    2              382.500000          191.250000
Residual      3               17.500000            5.833330

Total         5              400.000000           80.000000


F-Test Statistic      Value :     32.785733
       Degrees of Freedom 1 :      2
       Degrees of Freedom 2 :      3
                Probability :       .009151
+-----------------+
 AUTOCORRELATION 
+-----------------+

Durbin-Watson Statistic  :        2.914286
Von Neumann Ratio        :        3.642857

rho - Least Squares      :        -.740741
rho - Maximum Likelihood :        -.571429
rho - Serial Correlation :        -.714286
rho - Goldberger         :        -.650600

Number of Observations   :        5
+--------------------+
 MEAN AND VARIANCES 
+--------------------+

Variable y Observed                Mean :        8.000000
                      Unbiased Variance :       20.000000
                               St. Dev. :        4.472136
                        Biased Variance :       16.000000
                               St. Dev. :        4.000000

Variable y Calculated              Mean :        8.000000
                      Unbiased Variance :       15.625000
                               St. Dev. :        3.952847
                        Biased Variance :       12.500000
                               St. Dev. :        3.535534

Error                              Mean :         .000000
                      Unbiased Variance :        4.375000
                               St. Dev. :        2.091650
                        Biased Variance :        3.500000
                               St. Dev. :        1.870829

Number of Observations                  :        5
+--------------+
 CORRELATIONS 
+--------------+

Correlation  Y Observed   vs. Y Calculated :         .883883
             Y Observed   vs. Errors       :         .467707
             Y Calculated vs. Errors       :         .000000

Multiple Correlation Coefficient           :         .883883

Coefficient of Determination  Non Adjusted :         .781250
                                  Adjust.1 :         .708333
                                  Adjust.2 :         .718274

Number of Observations                     :        5
Number of Explanatory Variables            :        2
+-----------------+
 MODEL SELECTION 
+-----------------+

Akaike       Final Prediction Error        FPE :        8.166667
             Information Criterion         AIC :        7.789393
             Information Criterion      ln AIC :        2.052763
Amemiya      Prediction Criterion          APC :        8.166667
Craven-Wahba Generalized Cross Validation  GCV :        9.722222
Hannan-Quinn Criterion                     HQC :        5.121621
Rice         Criterion                      RC :       17.500000
Schwartz     Criterion                      SC :        6.662789
             Criterion                   ln SC :        1.896538
Shibata      Criterion                     SHC :        6.300000

Error Biased Variance                          :        3.500000
Log Likelihood Function                        :      -10.226600
Multiple Correlation Coefficient               :         .883883
Coefficient of Determination      Non Adjusted :         .781250
                                      Adjust.1 :         .708333
                                      Adjust.2 :         .718274

Number of Observations                         :        5
Number of Explanatory Variables                :        2
+-------------------+
 MODEL PERFORMANCE 
+-------------------+

Sum Squared Error                     SSE :       17.500000
Mean Squared Error                    MSE :        3.500000
Root Mean Squared Error               RMS :        1.870829
Error Biased Variance                     :        3.500000
Variance of Estimate                      :        5.833333
Standard Error of Estimate                :        2.415229
Log Likelihood Function                   :      -10.226600
Multiple Correlation Coefficient          :         .883883
Coefficient of Determination Non Adjusted :         .781250
                                 Adjust.1 :         .708333
                                 Adjust.2 :         .718274
z-Transform of Correlation Coefficient    :        1.393248

F-Test                          Statistic :       10.714286
                    Degrees of Freedom d1 :        1
                                       d2 :        3
                              Probability :         .046662

Number of Observations                    :        5
Number of Explanatory Variables           :        2
+------------------+
 ERROR STATISTICS 
+------------------+

Error Mean                         ME :         .000000
      Squared Mean                MES :         .000000
      Variance Biased            VARE :        3.500000
      St. Dev. Biased                 :        1.870829
      Variance Unbiased               :        4.375000
      St. Dev. Unbiased               :        2.091650

Relative Error Standard Deviation     :         .233854
Mean Squared Error                MSE :        3.500000
Root Mean Squared Error           RMS :        1.870829
Relative Root Mean Squared Error RRMS :         .535069
Mean Percentage Error             MPE :        2.519320
Mean Absolute Error               MAE :        1.400000
Mean Absolute Percentage Error  MAPE1 :         .134935
                                MAPE2 :         .136190

Variation Coefficient             VC1 :       23.385359
                                  VC2 :  **************

Number of Observations                :        5
+----------------------+
 DECOMPOSITION OF MSE 
+----------------------+

Theil's Inequality Coefficient  TH :         .105752
                                IC :         .209165
                               IC2 :         .043750

Mean Squared Error             MSE :        3.500000

Proportion due to Bias          UM :         .000000
                  Variance      US :         .061637
                  Covariance    UC :         .938363

Proportion due to Bias          UM :         .000000
                  Regression    UR :         .000000
                  Disturbance   UD :        1.000000

Number of Observations             :        5

Output - Graphical

References

Barten, A.
Note on Unbiased Estimation of the Squared Multiple Correlation Coefficient
Statistica Neerlandica, vol. 16, no. 2, 1962, pp. 151-163.

Durbin, J. and Watson, G.
Testing for Serial Correlation in Least-Squares Regression I
Biometrika, vol. 37, 1950, pp. 409-428.

Durbin, J. and Watson, G.
Testing for Serial Correlation in Least-Squares Regression II
Biometrika, vol. 38, 1951, pp. 159-178.

Forecasts and Realizations
Central Planning Bureau Monograph no.10
Staatsdrukkerij, s-Gravenhage, 1965.

Goldberger, A.
Econometric Theory
John Wiley, London, 1964, pp. 197-198.

Jarque, C.M. and Bera, A.K.
A Test for Normality of Observations and Regression Residuals
International Statistical Review, vol. 55, 1987, pp. 163-172.

Judge, G. et al.
The Theory and Practice of Econometrics
John Wiley, New York, 2nd Ed., 1985, p. 242.

Malinvaud, E.
Statistical Methods of Econometrics
Studies in Mathematical and Managerial Economics VI
North-Holland, Amsterdam, 2nd Ed., 1970, p.520.

Press, W.H. et al.
Numerical Recipes in Pascal - The Art of Scientific Computing
Cambridge University Press, Cambridge, 1989, pp. 543-545.

Ramanathan, R.
Introductory Econometrics with Applications
Harcourt Brace Jovanovich, San Diego, 2nd Ed., 1992, p. 167.

Theil, H.
Econometric Forecasts and Policy
Contributions to Economic Analysis XV
North-Holland, Amsterdam, 2nd Ed., 1961, pp. 26-43.

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Home
Up
Central Tendency
Concentration
Box Plot
Density Trace
Frequency Table
Moments
Correlation
Partial Correlation
Quartiles
Histogram
Skew./Peaked.
Rank Correlation
Variability
Simple Regression
General Linear Model
Mean and Variances
Covariance
Correlation Coefficient
Least Squares
Parameter b(0)
Parameter b(1)
Response
Significance
Determination Coeff.
ANOVA
Residuals
Autocorrelation
Model Selection
Model Performance
Relationships
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