# CASIO

User´s Guide

## Statistical Calculations (SD, REG*) * fx-82MS/fx-85MS/fx-300MS/fx-350MS only

### Standard Deviation (SD)

Use the key to enter the SD Mode when you want to perform statistical calculations using standard deviation.

• (SD)

In the SD Mode and REG Mode, the key operates as the key.

fx-82MS/fx-85MS/fx-300MS/fx-350MS: Always start data input with (CLR)(Scl) to clear statistical memory.
fx-220 PLUS: Always start data input with (Scl) to clear statistical memory.

Input data using the key sequence shown below.

<x-data>

Input data is used to calculate values for n, Σx, Σx2, x-, σn and sx, which you can recall using the key operations noted nearby.

To recall Σx2 this type of value: Perform this key operation: fx-82MS/fx-85MS/ fx-300MS/fx-350MS fx-220 PLUS (S-SUM)(Σx2) (Σx2) (S-SUM)(Σx) (Σx) (S-SUM)(n) (n) (S-VAR)(x-) (x-) (S-VAR)(σx) (σx) (S-VAR)(sx) (sx)

Example: To calculate sx, σx, x-, n, Σx, and Σx2 for the following data : 55, 54, 51, 55, 53, 53, 54, 52
fx-82MS/fx-85MS/fx-300MS/fx-350MS:

• In the SD Mode:
(CLR)(Scl)(Stat clear)

55

Each time you press to register your input, the number of data input up to that point is indicated on the display (n value).

545155535452

Sample Standard Deviation (sx) = 1.407885953

• (S-VAR)(sx)
• 1.407885953

Population Standard Deviation (σx) = 1.316956719

• (S-VAR)(σx)
• 1.316956719

Arithmetic Mean (x-) = 53.375

• (S-VAR)(x-)
• 53.375

Number of Data (n) = 8

• (S-SUM)(n)
• 8.

Sum of Values (Σx) = 427

• (S-SUM)x)
• 427.

Sum of Squares of Values (Σx2) = 22805

• (S-SUM)x2)
• 22805.

fx-220 PLUS:

• In the SD Mode:
(Scl)(Stat clear)

55

Each time you press to register your input, the number of data input up to that point is indicated on the display (n value).

545155535452

Sample Standard Deviation (sx) = 1.407885953

• (sx)
• 1.407885953

Population Standard Deviation (σx) = 1.316956719

• (σx)
• 1.316956719

Arithmetic Mean (x-) = 53.375

• (x-)
• 53.375

Number of Data (n) = 8

• (n)
• 8.

Sum of Values (Σx) = 427

• x)
• 427.

Sum of Squares of Values (Σx2) = 22805

• x2)
• 22805.

Data Input Precautions

inputs the same data twice.

fx-82MS/fx-85MS/fx-300MS/fx-350MS: You can also input multiple entries of the same data using (;). To input the data 110 ten times, for example, press 110(;) 10.
fx-220 PLUS: You can also input multiple entries of the same data using (;). To input the data 110 ten times, for example, press 110(;) 10.

You can perform the above key operations in any order, and not necessarily that shown above.

While inputting data or after inputting data is complete, you can use the and keys to scroll through data you have input. If you input multiple entries of the same data using (;) ((;) on the fx-220 PLUS) to specify the data frequency (number of data items) as described above, scrolling through data shows both the data item and a separate screen for the data frequency (Freq).

You can then edit the displayed data, if you want. Input the new value and then press the key to replace the old value with the new one. This also means that if you want to perform some other operation (calculation, recall of statistical calculation results, etc.), you should always press the key first to exit data display.

Pressing the key instead of after changing a value on the display registers the value you input as a new data item, and leaves the old value as it is.

You can delete a data value displayed using and by pressing (CL). Deleting a data value causes all values following it to be shifted up.

Data values you register are normally stored in calculator memory. The message "Data Full" appears and you will not be able to input any more data if there is no memory left for data storage. If this happens, press the key to display the screen shown below.

Press to exit data input without registering the value you just input.

Press if you want to register the value you just input, without saving it in memory. If you do this, however, you will not be able to display or edit any of the data you have input.

To delete data you have just input, press (CL).

After inputting statistical data in the SD Mode or REG Mode, you will be unable to display or edit individual data items any longer after perform either the following operations.

Changing to another mode

Changing the regression type (Lin, Log, Exp, Pwr, Inv, Quad)

### Regression Calculations (REG) (fx-82MS/fx-85MS/fx-300MS/fx-350MS only)

Use the key to enter the REG Mode when you want to perform statistical calculations using regression.

• (REG)

In the SD Mode and REG Mode, the key operates as the key.

Entering the REG Mode displays screens like the ones shown below.

Press the number key (, , or ) that corresponds to the type of regression you want to use.

(Lin) : Linear regression

(Log) : Logarithmic regression

(Exp) : Exponential regression

(Pwr) : Power regression

(Inv) : Inverse regression

Always start data input with (CLR)(Scl) to clear statistical memory.

Input data using the key sequence shown below.

<x-data><y-data>

The values produced by a regression calculation depend on the values input, and results can be recalled using the key operations shown in the table below.

To recall this type of value: Perform this key operation: (S-SUM)(Σx2) (S-SUM)(Σx) (S-SUM)(n) (S-SUM)(Σy2) (S-SUM)(Σy) (S-SUM)(Σxy) (S-VAR)(x-) (S-VAR)(σx) (S-VAR)(sx) (S-VAR)(y-) (S-VAR)(σy) (S-VAR)(sy) (S-VAR)(A) (S-VAR)(B) (S-VAR)(r) (S-VAR)(xˆ) (S-VAR)(yˆ)

The following table shows the key operations you should use to recall results in the case of quadratic regression.

To recall this type of value: Perform this key operation: (S-SUM)(Σx3) (S-SUM)(Σx2y) (S-SUM)(Σx4) (S-VAR)(C) (S-VAR)(xˆ1) (S-VAR)(xˆ2) (S-VAR)(yˆ)

The values in the above tables can be used inside of expressions the same way you use variables.

Linear Regression

The regression formula for linear regression is: y = A + Bx.

Example: Atmospheric Pressure vs. Temperature

Perform linear regression to determine the regression formula terms and correlation coefficient for the data below.

Temperature Atmospheric Pressure 1003 hPa 1005 hPa 1010 hPa 1011 hPa 1014 hPa

Next, use the regression formula to estimate atmospheric pressure at -5°C and temperature at 1000 hPa. Finally, calculate the coefficient of determination (r2) and sample covariance (xy - nx-y-n - 1).

• In the REG Mode:
(Lin)
(CLR)(Scl)(Stat clear)
101003

Each time you press to register your input, the number of data input up to that point is indicated on the display (n value).

151005201010251011301014

Regression Coefficient A = 997.4

• (S-VAR)(A)
• 997.4

Regression Coefficient B = 0.56

• (S-VAR)(B)
• 0.56

Correlation Coefficient r = 0.982607368

• (S-VAR)(r)
• 0.982607368

Atmospheric Pressure at 5°C = 994.6

• 5(S-VAR)(yˆ)
• 994.6

Temperature at 1000 hPa = 4.642857143

• 1000(S-VAR)(xˆ)
• 4.642857143

Coefficient of Determination = 0.965517241

• (S-VAR)(r)
• 0.965517241

Sample Covariance = 35

• (S-SUM)xy)
(S-SUM)(n)
(S-VAR)(x-)
(S-VAR)(y-)
(S-SUM)(n)1
• 35.

Logarithmic, Exponential, Power, and Inverse Regression

Use the same key operations as linear regression to recall results for these types of regression.

The following shows the regression formulas for each type of regression.

Logarithmic Regression y = A + B･ln x y = A･eB•x (ln y = ln A + Bx) y = A･xB (ln y = ln A + Bln x) y = A + B･1/x

The regression formula for quadratic regression is: y = A + Bx + Cx2.

Example:

Perform quadratic regression to determine the regression formula terms for the data below.

xi yi 1.6 23.5 38.0 46.4 48.0

Next, use the regression formula to estimate the values for yˆ (estimated value of y) for xi = 16 and xˆ (estimated value of x) for yi = 20.

In the REG Mode:
(CLR)(Scl)(Stat clear)

29165023574380103464118480

Regression Coefficient A = -35.59856934

• (S-VAR)(A)
• -35.59856934

Regression Coefficient B = 1.495939413

• (S-VAR)(B)
• 1.495939413

Regression Coefficient C = -6.71629667 × 10-3

• (S-VAR)(C)
• -6.71629667×10-3

yˆ when xi is 16 = -13.38291067

• 16(S-VAR)(yˆ)
• -13.38291067

xˆ1 when yi is 20 = 47.14556728

• 20(S-VAR)(xˆ1)
• 47.14556728

xˆ2 when yi is 20 = 175.5872105

• 20(S-VAR)(xˆ2)
• 175.5872105

Data Input Precautions

inputs the same data twice.

You can also input multiple entries of the same data using (;). To input the data "20 and 30" five times, for example, press 20 30 (;) 5.

The above results can be obtained in any order, and not necessarily that shown above.

Precautions when editing data input for standard deviation also apply for regression calculations.

Do not use variables A through F, X, or Y to store data when performing statistical calculations. These variables are used for statistical calculation temporary memory, so any data you may have assigned to them may be replaced by other values during statistical calculations.

Entering the REG Mode and selecting a regression type (Lin, Log, Exp, Pwr, Inv, Quad) clear variables A through F, X, and Y. Changing from one regression type to another inside the REG Mode also clears these variables.