How To Use The Least Squares Regression Calculator
This is a online regression calculator for statistical use.
Enter your data as a string of number pairs, separated by
commas. Enter each data point as a separate line. Then hit
calculate. The linear regression calculator will estimate
the slope and intercept of a trendline that is the best fit
with your data.
This page includes a regression equation calculator, which
will generate the parameters of the line for your analysis.
It can serve as a slope of regression line calculator,
measuring the relationship between the two factors. This
tool can also serve as a sum of squared residuals calculator
to give you a perspective on fit & accuracy.
You can save your data for use with this webpage and the
similar tools on this site. Just hit the "save data"
button. It will save the data in your browser (not on our
server, it remains private). It will appear on the list of
saved datasets below the data entry panel. To retrieve it,
all you need to do is click the "load data" button next to it.
Interpreting The Least Squares Regression Calculator Results
This linear regression calculator fits a trend-line to your data using the
least squares technique. This approach optimizes the fit of the
trend-line to your data, seeking to avoid large gaps between the
predicted value of the dependent variable and the actual value.
The Least Squares Regression Calculator will return the slope of the line and the y-intercept.
It will also generate an R-squared statistic, which evaluates how
closely variation in the independent variable matches variation in the
dependent variable (the outcome). For a deeper view of the mathematics
behind the approach, here's a regression tutorial.
To help you visualize the trend - we display a plot of the
data and the trend-line we fit through it. If you hover or tap on
the chart (in most browsers), you can get a predicted Y value for
that specific value of X.The equation of the line is of
particular interest since you can use it to predict points
outside your original data set. Similarly, the r-squared gives
you an estimate of the error associated with effort: how far
the points are from the calculated least squares regression line.
Real World Analysis
Some practical comments on real world analysis:
The modeling process only looks at the mean of the
dependent variable. This is important if you're concerned with a small subset of the population, where extreme values trigger extreme outcomes.
Data observations must be truly independent. Each observation in the
model must truly stand on its own. Two common pitfalls - space and
time. The first - clustering in the same space - is a function of
convenience sampling. The model can't predict behavior it cannot see
and assumes the sample is representative of the total population. If
you attempt to use the model on populations outside the training set,
you risk stumbling across unrepresented (or under-represented) groups.
Clustering across time is another pitfall - where you re-measure the
same individual multiple times (for medical studies). Both of these
can bias the training sample away from the true population dynamics.
- Using a linear model assumes the underlying process
you are modeling behaves according to a linear system. This is often
not the case; many engineering and social systems are driven by different dynamics better represented by exponential, polynomial, or power models.
Keep this in mind when you use the Least Squares Regression Calculator - are you fitting the correct curve?
The R-squared metric isn't perfect, but can alert you to when you are
trying too hard to fit a model to a pre-conceived trend.
On the same note, the linear regression process is very sensitive to
outliers. The Least Squares Regression Calculator is biased against data points
which are located significantly away from the projected trend-line.
These outliers can change the slope of the line disproportionately.
On a similar note, use of any model implies the underlying process has
remained 'stationary' and unchanging during the sample period. If there
has been a fundamental change in the system, where the underlying rules
have changes, the model is invalid. For example, the risk of employee
defection varies sharply between passive (happy) employees and agitated (angry) employees who are shopping for a new opportunity.
The underlying calculations and output are consistent with most statistics
packages. It applies the method of least squares to fit a line through your
data points. The equation of the regression line is calculated, including
the slope of the regression line and the intercept. We also include the
r-square statistic as a measure of goodness of fit. This equation can be
used as a trendline for forecasting (and is plotted on the graph).
Want to know more? This page has some handy linear regression resources.