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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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