# 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:

## 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.

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 x

_{1}, x_{2}, ..., x_{n}occurring with weights y_{1}, y_{2}, ..., y_{n}. To calculate weighted mean:- Use
**SUM+**to enter the data as x,y-pairs. The y-values are the weights of the x-values. - 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). |

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