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HP 20s Calculator - Mean, Standard Deviation, and Summation Statistics

Introduction
The HP 20s calculates the mean, standard deviation, n, sum, and sum of the squares of x-data. For y-data, it calculates the mean and standard deviation, sum, and sum of the squares. The HP 20s also calculates the sum of the products of x- and y- data.
Calculator symbol key
The procedures in this document use the following text to represent symbol keys:
Key
Description
Text Representation
Right shift key
RS
Left-shift key
LS
Divide
divide
Accumulates statistical data
SUM+
Arithmetic mean (average) of the x-values
xy
Square root
SQRT
Square root key
SQx
Mean of the x-values weighted by the y-values
xw
Example of calculating the mean, standard deviation, and root mean square
A yacht captain wants to determine how long it takes to change a sail. She randomly chooses six members of her crew, observes them as they carry out the sail change, and records the number of minutes required: 4.5, 4, 2, 3.25, 3.5, 3.75.
Calculate the mean and standard deviation of the times. Also, calculate the root mean square, using the formula in Figure 1.
Figure : Formula for calculating the root mean square
Keys
Display
Description
Press RS, then CLSUM
0.0000
Clears the statistical registers
Press 4.5, then SUM+
1.0000
Enters the first time
Press 4, then SUM+
2.0000
Press 2, then SUM+
3.0000
Press 3.25, then SUM+
4.0000
Press 3.5, then SUM+
5.0000
Press 3.75, then SUM+
6.0000
Press RS, then xy
3.5000
Calculates the mean
Press RS, then Sx,Sy
0.8515
Calculates the standard deviation
Press RCL, then 7
77.1250
Displays the sum of squares of x-data
Press divide, RCL, then 4
6.0000
Displays n
Press =, then SQx
3.5853
Calculates the root mean square
Calculating the population standard deviation
The standard deviations calculated by entering RS, Sx,Sy, RS, Sx,Sy, LS, then SWAP are the sample standard deviations. They assume that the data is a sampling of a larger, complete set of data. If the data constitutes the entire population of data, the true population standard deviation can be calculated by calculating the mean of the original data, adding the mean to the statistical data using SUM+, and then calculating the standard deviation. For two-variable statistics, after calculating the mean of the original data, press LS, then SWAP to put the data in the proper order (mean average of the y-values in the display), then press SUM+
Example of calculating population standard deviation
The coach has four new players on the team with heights of 193, 182, 177, and 185 centimeters and weights of 90, 81, 83, and 77 kilograms. Find the mean and population standard deviations of their heights and weights.
Keys
Display
Description
Press RS, then CLSUM
0.0000
Clears the statistical registers.
Press 193, INPUT, 90, then SUM+
1.0000
Enters height and weight of player 1.
Press 182, INPUT, 81, then SUM+
2.0000
Enters height and weight of player 2.
Press 177, INPUT, 83, SUM+
3.0000
Enters height and weight of player 3.
Press 185, INPUT, 77, SUM+
4.0000
Enters height and weight of player 4.
Press RS, then xy
184.2500
Calculates mean of heights (x).
Press LS, then SWAP
82.7500
Displays mean of weights (y).
Press SUM+
5.0000
Adds means to data. (Data must be in x, y order with the y in the display).
Press RS, then SxSy
5.8041
Calculates population standard deviation for heights (x).
Press LS, then SWAP
4.7104
Displays population standard deviation for weights (y).
Weighted mean
An example of weighted mean is x1, x2, ..., xn occurring with weights y1, y2, ..., yn. To calculate weighted mean:
  1. Use SUM+ to enter the data as x,y-pairs. The y-values are the weights of the x-values.
  2. Press RS, then xw
Example of calculating weighted mean
A manufacturing company purchases a certain part four times a year. Last year's purchases were:
Price/Part
$4.25
$4.60
$4.70
$4.10
Number of parts
250
800
900
1000
Calculate the average price per part.
Keys
Display
Description
Press RS, then CLSUM
0.0000
Clears the statistical registers.
Press 4.25, INPUT, 250, then SUM+
1.0000
Press 4.6, INPUT, 800, then SUM+
2.0000
Press 4.7, INPUT, 900, then SUM+
3.0000
Press 4.1, INPUT, 1000, then SUM+
4.0000
Enters the data and their weights.
Press RS, then xw
4.4314
Calculates weighted mean (average part price).
Figure : Statistical formulas

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