Statistical Functions

COEF_A (Regression Coefficient)

Calculates the coefficient A of the current regression mode formula. This function is accessible from the [S-DIST] key when the calculator is in one of the regression modes.

COEF_A accepts no input value, and the result is applicable to the regression formula used, as follows:

For an example calculation using regression, see the Statistical Calculations section. See also the COEF_B, COEF_C and COEF_D.

COEF_B (Regression Coefficient)

Calculates the coefficient B of the current regression mode formula. This function is accessible from the [S-DIST] key when the calculator is in one of the regression modes. COEF_B accepts no input value. See COEF_A for more information.

COEF_C (Regression Coefficient)

Calculates the coefficient C for those regression modes where a third coefficient is used, including quadratic, cubic and logistic regression. This function is accessible from the [S-DIST] key. COEF_C accepts no input value. See COEF_A for more information.

COEF_D (Regression Coefficient)

Calculates the coefficient D for those regression modes where a forth coefficient is used, namely cubic regression. This function is accessible from the [S-DIST] key. COEF_D accepts no input value. See COEF_A for more information.

COEF_R (Correlation)

Calculates the correlation coefficient R for linear and regression modes supporting a linear transform model. This function is accessible from the [S-DIST] key when the calculator is in one of these regression modes.

A correlation coefficient is a number between -1 and +1, which expresses the degree to which X and Y variables are related, either by a direct linear relationship or when the values are linearized by the transform model. If there is a perfect relationship with a positive slope, COEF_R returns +1. If there is a perfect relationship with negative slope, COEF_R returns -1. A correlation coefficient of 0 means that there is no relationship between the variables.

COEF_R accepts no input value. See also the RMSE, COEF_R2 and COVAR functions.

COEF_R2 (Determination)

Calculates the coefficient of determination for linear and regression modes supporting a linear transform model. This function is accessible from the [S-DIST] key when the calculator is in one of these regression modes.

The coefficient of determination is simply the square of the correlation coefficient COEF_R and is, thus, an unsigned value between 0 and +1 representing the 'goodness of fit'. Values greater than 0.64 are often considered to represent a good regression fit.

Copy RegEq

Simply copies the regression formula to the Window's clipboard as a text string; it does not perform any mathematical operation. This operation is accessible from the [S-DIST] key. It is not available in SD mode.

Example

In Quadratic Regression mode press:

[S-DIST] and select Copy RegEq
The clipboard should now contain, "y = Ax^2 + Bx + C".

COVAR

Calculates the covariance between the X and Y for linear and regression modes supporting a linear transform model. This function is accessible from the [S-DIST] key when the calculator is in one of these regression modes.

COVAR provides a measure of the extent to which the X and Y values co-vary. Because the number representing covariance depends on the units of the data, it is often difficult to compare covariances among data sets of different scales. The correlation functions COEF_R and COEF_R2 address this problem.

DAT (Input Key)

Pressing [DAT] appends the display value to the current statistical list, thus allowing the input of data via the main keypad.

To enter a value, simply key it in and press [DAT]. You should use the separator key [;] to enter paired values, for example:

45 [;] 3
[DAT]

In SD mode, this will enter a value of 45 with a frequency of 3, or an X-Y pair in regression modes. If you omit the frequency in SD mode, a value of 1 will be assumed. For large amounts of data, you will find it convenient to use the Graphing & List Window instead.

Note that [;] is also used to enter complex numbers, however, the input does not become complex until a calculation is performed.

zs(r) (Z-Score)

Returns the two-sided z-value, sometimes known as the standard score or z-score, given a confidence level input in the range (-1, 1). This function is found on the main keypad, and is independent of the statistical mode and data list.

A z-value is equal to a Student t-value for an infinite degree of freedom, and can be used as an approximate of a t-value provided it is used in connection with a calculation using a large sample (typically above 30).

For example, provided the number of values used to calculate StdErrX is considered large, the following input gives the confidence interval of the mean:

[zs] (z-score function above the [;] key)
0.9 (90% confidence as fraction)
[×]
[S-VAR] and select StdErrX
[ENTER]
Displays: 0.21161705529476782 (result depends on SD list contents)

In general, if the number of samples under consideration is less than 30, the z-value should not be used.

It is common to define the confidence level in terms of an alpha value, or risk of error. In this case, r = (1 - alpha), as follows:

zs(r) = zs(1 - alpha)

and examples include:

zs(0.90) = zs(1 - 0.1) = 1.6449 (90% confidence, or 10% risk)
zs(0.95) = zs(1 - 0.05) = 1.96 (95% confidence, or 5% risk)
zs(0.99) = zs(1 - 0.01) = 2.5758 (99% confidence, or 1% risk)

Note that, where inputs values are close to -1 or +1 (ie. 0.99999999999), the result will be considered to approximate infinity and DreamCalc with give "range error". Otherwise, the result is accurate to the number of digits shown.

The plot below shows zs(r) over the input range (-1, 1).

Where a one-sided z-value is required, the following simple relation can be applied:

zone-sided = zs(2*r - 1)
For example, we can calculate a one-sided 90% z-score using prefix algebraic input, as follows:

[zs]
[(]
2
[×]
0.9 (90% confidence)
[-]
1
[)]
[ENTER]
Displays: 1.2816 (one-sided result for 90% confidence)

You may also wish to note that you can perform the reverse of zs(r) using the DreamCalc QG(z), by way of the following relation:

2 * QG( zs(r) ) = r (two-sided interval)
and the following is applicable for a one-sided interval:

PG( zs(2*r - 1) ) = r (one-sided interval)