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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.

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

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.

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

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.

FX(x)

Calculates y, as a function x, where:

y = f(x)

using the regression formula determined for the current X-Y data.

This function is accessible from the [S-DIST] key, and is available only in X-Y modes.

Example:

[S-DIST] and select FX(x)
55
[ENTER]
Displays: 132.7733

See also the reverse FY(y) function, which calculates x as a function of y. See also Copy RegEq, and COEF_A to COEF_D.

FY(y)

Calculates x, as a function y, where:

x = f(y)

using the regression formula determined for the current X-Y data. Where multiple solutions exist for x, only one result is returned.

This function is accessible from the [S-DIST] key, and is available only in X-Y modes. See also FX(x), Copy RegEq, and COEF_A to COEF_D.

Gk (Kurtosis)

Calculates the kurtosis Gk of the SD list data. This function is accessible from the [S-DIST] key when the calculator is in SD statistical mode. It is not available in X-Y regression modes.

Kurtosis is a measure of the "peakedness" of the data, and is defined by:

Gk = ∑(x_n - MeanX)⁴ / (StdX[N-1]⁴ * (N-1))

Positive kurtosis indicates a "peaked" distribution and negative kurtosis indicates a "flat" distribution. A normal distribution of data has a kurtosis value of +3.

Gk accepts no input value. See also: Gs (skew).

GRan# (Gaussian Random)

Returns a random number with a standard normal distribution, ie. with a mean of zero and standard deviation of one. GRan# accepts no input value.

Unlike other statistical functions, GRan# is to be found under the MISC menu key.

Gs (Skew)

Calculates the level of skew Gs of the SD list data. This function is accessible from the [S-DIST] key when the calculator is in SD statistical mode. It is not available in X-Y regression modes.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. It is defined by:

Gs = ∑(x_n - MeanX)³ / (StdX[N-1]³ * (N-1))

Positive skew indicates that the data distribution is skewed right. By skewed right, we mean that the right tail is long relative to the left tail. Whereas a negative value indicates data that are skewed left. The skewness of symmetric data, including a normal distribution, is zero.

Gs accepts no input value. See also: Gk (kurtosis).

MaxX

Returns the maximum X value in the list. This function is accessible from the [S-SUM] key. It can be used in either SD or X-Y modes. MaxX accepts no input value. See also MinX and RangeX.

MaxY

Returns the maximum Y value in the regression data list. This function is accessible from the [S-SUM] key. It is not available in SD mode. MaxY accepts no input value.

MeanX

Returns the calculated mean of the X values in the list. This function is accessible from the [S-VAR] key. It can be used in either SD or X-Y modes. In SD mode, account is taken of frequency values. MeanX accepts no input value.

MeanY

Returns the calculated mean of the Y values in the regression data list. This function is accessible from the [S-VAR] key. It is not available in SD mode. MeanY accepts no input value.

MedianX

Returns the median value, calculated for the X values in the list. MedianX is accessible from the [S-DIST] key in SD mode only.

The median is the value which is half-way between the minumum and maximum values--the mid-value if the data were to be sorted.

MedianX and accepts no input value. See also ModeX.

MinX

Returns the minimum X value in the list. This function is accessible from the [S-SUM] key. It can be used in either SD or X-Y mode. MinX accepts no input value. See also MaxX and RangeX.

MinY

Returns the minimum Y value in the regression data list. This function is accessible from the [S-SUM] key. It is not applicable in SD mode. MinY accepts no input value.

ModeX

Returns the most frequent value in the list of discrete data. ModeX is accessible from the [S-DIST] key in SD mode only.

For example, given the values 1, 2, 3, 3, 4, with all frequencies 1, the result of ModeX will be 3 as this value appears twice. If no most frequent value exists, a math error will be given. For example, the values 1, 2, 3, 4 will give an error because no single value appears more than any other.

ModeX accepts no input value. See also MedianX.

N (Sum of Frequency)

Returns the sum of the frequency values in the list. The N function is accessible from the [S-SUM] key. It can be used in either SD or X-Y regression modes.

In SD mode, the result is the sum of frequencies. If all frequency values are 1, the result is simply the number of data values. In X-Y modes, the result is always the data count.

The N function accepts no input value.

P_G (Gaussian Probability)

Given an input value z, the P_G function returns the integral of the standard Gaussian curve over the range [-INF, z]. This function is found on the main keypad, and is independent of the statistical mode and data list.

The result is the area shown under the curve below, and represents the probability for the range -INF to z, where z is expressed in terms of a normalized variate (the number of standard deviations from the mean).

Gaussian Probability P_G

Example:

Given the following sample data, estimate the probability of observing sample values less than 8.8. Assume that the data is normally distributed.

10.88
10.25
10.2
8.71
11.23
10.88
12.33
10.67
9.96
9.08
8.76

9.23
10.56
11.09
10.7
11.08
9.84
9.59
8.22
9.89
10.6
10.61

9.12
9.6
10.58
9.11
10.02
10.51
10.96
11.11
9.85
8.95
10.16

10.11
9.39
9.28
10.27
9.52
9.1
10.1
9.65
8.48
9.47
10

Ensure that DreamCalc is in SD statistical mode, and key in the above data.

In prefix algebraic input mode, enter:

[PG] (probability from -INF to z)
[S-DIST] and select Z (normalized variate)
8.8
[ENTER]
Displays: 0.081034 (probability result)

DreamCalc employs a numerical integration technique to calculate P_G. Input values of -5 or less will give 0, while inputs of +5 or more will return +1. Otherwise, the result is accurate to the number of digits shown.

Q_G (Gaussian Probability)

Given an input value z, the Q_G function returns the integral of the standard Gaussian curve over the range [0, z]. This function is found on the main keypad, and is independent of the statistical mode and data list.

The result is the area shown under the curve below, and represents the probability for the range 0 to z, where z is expressed in terms of a normalized variate (the number of standard deviations from the mean).

Gaussian Probability Q_G

DreamCalc employs a numerical integration technique to calculate Q_G. Input values of -5 or less, or +5 or more, will both give 0.5. Otherwise, the result is accurate to the number of digits shown.

R_G (Gaussian Probability)

Given an input value z, the R_G function returns the integral of the standard Gaussian curve over the range [z, +INF]. This function is found on the main keypad, and is independent of the statistical mode and data list.

The result is the area shown under the curve below, and represents the probability for the range z to +INF, where z is expressed in terms of a normalized variate (the number of standard deviations from the mean).

Gaussian Probability R_G

DreamCalc employs a numerical integration technique to calculate R_G. Input values of -5 or less will give +1, while inputs of +5 or more will return 0. Otherwise, the result is accurate to the number of digits shown.

RangeX

Returns the difference between the maximum and minimum X values in the list. This function is accessible from the [S-SUM] key. It can be used in either SD or X-Y regression modes. RangeX accepts no input value. See also MaxX and MinX.

RangeY

Returns the difference between the maximum and minimum Y values in the X-Y data list. This function is accessible from the [S-SUM] key. It is not applicable in SD mode. RangeY accepts no input value. See also MaxY and MinY.

S-DIST (Menu Key)

The [S-DIST] key provides input, via an on-screen menu, of functions related to distribution calculations.

For SD mode, functions include: Z, ModeX, Median, Skew Gs and Kurtosis Gk.

For X-Y modes, functions include: coefficients, functions FX(x) and FY(y), RMSE, covariance etc.

Note. Press this key repeatedly times to page through all available functions.

RMSE

Calculates the root mean square error of y in the X-Y list. This function is accessible from the [S-DIST] key and can be used in X-Y modes only.

The RMSE is the distance, on average, of data points from the line of best fit, measured along the vertical. It provides a measure of the error of fit in the same units as the y-axis. Unlike COEF_R and COEF_R², RMSE can be used in all regression modes, including quadratic, cubic and logistic regression.

StdX[N]

Returns the population standard deviation of the values in the list, as defined by:

StdX[N] = √( ∑(x_n - MeanX)² / N )

Standard deviation is a measure of how widely spread the values in of data set are. For normally distributed data, one standard deviation equates to a range around the mean where the probability of observing a value is approximately 68.27%.

This function is accessible from the [S-VAR] key and can be used both SD and X-Y modes. In SD mode, account is also taken of frequency values. StdX[N] accepts no input value. See also the StdX[N-1] function.

StdY[N]

Returns the calculated population standard deviation of the Y values in the X-Y list. StdY[N] accepts no input value. This function is accessible from the [S-VAR] key. It is applicable in X-Y modes only. See also StdX[N].

StdX[N-1]

Returns the sample standard deviation of the values in the list, as defined by:

StdX[N] = √( ∑(x_n - MeanX)² / (N-1) )

Standard deviation is a measure of how widely spread the values in of data set are. For normally distributed data, one standard deviation equates to a range around the mean where the probability of observing a value is approximately 68.27%.

This function is accessible from the [S-VAR] key and can be used both SD and X-Y modes. In SD mode, account is also taken of frequency values. StdX[N-1] accepts no input value. See also the StdX[N] function.

StdY[N-1]

Returns the calculated sample standard deviation of the Y values in the X-Y list. StdY[N-1] accepts no input value. This function is accessible from the [S-VAR] key. It is applicable in X-Y modes only. See also StdX[N-1].

StdErrX

Returns the calculated standard error of the mean of values in the list, as defined by:

StdErrX = StdX[N-1] / √N

The standard error of the mean describes the probable error associated with the sample mean. It is a measure of how close the sample mean is likely be to the true or population mean.

This function is accessible from the [S-VAR] key and can be used both SD and X-Y modes. In SD mode, account is also taken of frequency values. StdX[N-1] accepts no input value.

StdErrY

Returns the calculated standard error of the Y values in the X-Y list. This function is accessible from the [S-VAR] key. It is applicable in X-Y modes only. See also StdErrX.

S-SUM (Menu Key)

The [S-SUM] key provides input, via an on-screen menu, of functions related to sums and ranges of the data.

Functions include: SumX, SumX², MinX, MinX, and RangeX etc. Equivalent functions are available for Y value calculations in X-Y modes.

Note. Press this key repeatedly times to page through all available functions.

SumX

Returns the sum of all x values in the list. This function is accessible from the [S-SUM] key, and can be used in either SD or X-Y mode. In SD mode, account is taken of frequency values. SumX accepts no input value.

SumX²

Returns the sum of the squares of x values in the list. This function is accessible from the [S-SUM] key. It can be used in either SD or X-Y mode. In SD mode, account is taken of frequency values. SumX² accepts no input value.

SumXY

Returns the sum of the x-y products in the regression data list. This function is accessible from the [S-SUM] key. It is not available in SD mode. SumXY accepts no input value.

SumY

Returns the sum of all y values in the regression data list. This function is accessible from the [S-SUM] key. It is not available in SD mode. SumY accepts no input value.

SumY²

Returns the sum of the squares of y values in the regression data list. This function is accessible from the [S-SUM] key. It is not available in SD mode. SumY² accepts no input value.

S-VAR (Menu Key)

The [S-KEY] key provides input, via an on-screen menu, of functions related to variance calculations.

Functions include: MeanX, StdX[N], StdX[N-1], StdErrX. Equivalent functions are available for Y value calculations in X-Y modes. Variance can be calculated by simply squaring the result of applicable standard deviation function.

Note. Press this key repeatedly times to page through all available functions.

z_s(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)

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

z_1-sided = z_s(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)

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

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.

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

2 * Q_G( z_s(r) ) = r (two-sided interval)

and the following is applicable for a one-sided interval:

P_G( z_s(2*r - 1) ) = r (one-sided interval)

See also Q_G, P_G and Z.

Z (Normalized Variate)

Calculates the normalized variate Z given an input value, using the data in the SD list. This function is accessible from the [S-DIST] key when the calculator is in SD statistical mode. It is not available in X-Y regression modes.

The normalized variate is the number of standard deviations between the mean and the input value x, as calculated with the following equation:

Z = (x - MeanX) / StdX[N-1]

The calculation assumes that the data is normally distributed. See the P_G function for an example of a calculation using the normalized variate.