# Beta version

BETA TEST VERSION OF THIS ITEM

This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

# Calculation data - measurement points

x-values | ||

y-values |

# Results - linear approximation of your points

Approximation | ||

Function formula | Show source$y=2~x+1$ | |

Line slope a | 2 | |

Free term b | 1 | |

Measurement points | ||

Number of points | 4 | |

Points you entered | (1, 3), (2, 5), (3, 7), (4, 9) | |

Helper values | ||

Sum of x-values $\sum x$ | 10 | |

Sum of y-values $\sum x$ | 24 | |

Sum of x squares $\sum x^2$ | 30 | |

Sum of multiplies $\sum xy$ | 70 |

# Some facts

- Approximation of a function consists in finding a
**function formula**that**best matches**to a**set of points**e.g. obtained as measurement data. - If we limit the search to
**linear function**only, then we say about**linear regression**or**linear approximation**. - The
**least squares method**is one of the methods for**finding such a function**. - The least squares method is the
**optimization method**. As a result we get function that the**sum of squares of deviations from the measured data is the smallest**. Mathematically, we can write it as follows:

$\sum_{i=1}^{n} \left[y_i - f(x_i)\right]^2 = min.$where:

**$(x_i, y_i)$**- coordinations of the i-th measurement point, these are**points that we know**,

**$f(x)$**- the function we are**searching for**, we want this function to best match to the measurement points,

**$n$**- number of measurement points.

- If we set a condition that we are only looking for a linear function:

$f(x) = ax + b$we get following solution:

$a = \frac{n~S_{xy} - S_x~S_y}{n~S_{xx} - \left(S_x\right)^2}$$b = \frac{S_y - a~S_x}{n}$where:

**$S_{x}$**- sum of x-values $\sum x_i$,

**$S_{y}$**- sum of y-values $\sum y_i$,

**$S_{xx}$**- sum of squares $\sum x_i^2$,

**$S_{xy}$**- sum of multiplies $\sum x_i~y_i$.

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