# Descriptive Statistics - Simple Linear Regression - Example

#### Free Statistics Calculator

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

#### Data

Example Data Set & Computation (°)

#### 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

#### 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

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

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
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

#### 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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