BrownMath.com→TI-83/84/89→SampleStatistics

Updated20Jan2021(What’sNew?)

Copyright © 2007–2024 by StanBrown, BrownMath.com

**Summary:**You can use your TI-83/84 to find**measures of central tendency and measures of dispersion**for a sample.

**Contents:**

- Descriptive Statistics for a List of Numbers
- Step 1: Enter the numbers in L1.
- Step 2: Compute the statistics.
- Step 3: Find the variance.

- Descriptive Statistics for a Frequency Distribution
- Step 1: Enter class marks in L1 and frequencies in L2.
- Step 2: Compute the statistics.
- Step 3: Find the variance.

- What’s New?

**Seealso:**MATH200A Program— Basic Statistics Utilities forTI-83/84 gives a downloadable programto plot histograms and box-whisker diagrams.

**Seealso:**optional advanced material: MATH200B Program part1gives a downloadable program that computes skewness and kurtosis, twonumerical measures of shape

## Descriptive Statistics for a List of Numbers

Quiz scores in a (fictitious) class were10.5, 13.5, 8, 12, 11.3, 9, 9.5, 5,15, 2.5, 10.5, 7, 11.5, 10, and 10.5.It’s hard to get much of a senseof the class by just staring at the numbers, butyou can easily compute the common measures of center and spread byusing your TI-83 or TI-84.

### Step 1: Enter the numbers in L1.

By the way, this note uses list L1, but you can actually use anylist you like, as long as you enter the actual list name in the`1-VarStats`

command in Step 2.(It doesn’t matter whether there are numbers in any other list.)

Enter the data points. | [`STAT` ] [`1` ] selects the list-edit screen. Cursor onto the label `L1` at top of first column, then [`CLEAR` ] [`ENTER` ] erases the list. Enter the x values. |

### Step 2: Compute the statistics.

Select the `1-VarStats` command. | [`STAT` ] [`►` ] [`1` ] pastes the command to the home screen. |

Specify which statistics list contains the data set. Show your work: write down `1-VarStats` and the list name. | Assuming you used `L1` , enter [`2nd` `1` makes `L1` ]. Press [ `ENTER` ] to execute the command. |

The important statistics are

**sample size***n*=15

Always check this first to guard against leaving out numbers orentering numbers twice.**mean***x̅*=9.72

(Write down symbol μ instead of*x̅*if this is a population mean.)**standard deviation***s*=3.17

Since this data set is a sample, use`Sx`

and write*s*for thestandard deviation. (When the data set is the whole population, use`σx`

and write σ for the standard deviation.)

If rounding is necessary, remember that we**round mean and standard deviation to one decimal place more than the data.****variance**is not shown on this screen; seeStep 3 below.

The down arrow on the screen tells you that there’s more information if you scroll down— in this case it’s the five-number summary. | [`▼` 5times ] for the five-number summary. |

You can tell the shape of the distribution. Sincethe mean *x̅*= 9.72 is just a hair less than the median`Med`

or *x̃*= 10.5, you know that the distribution is**slightly skewed left**.

The **range** is`Xmax`

− `Xmin`

= 15− 2.5=12.5.

The **interquartile range or IQR**is `Q3`

− `Q1`

= 11.5− 8=3.5. Recallthat we use 1.5× IQR toclassify outliers: we call a data point an outlier if it’sat least that far below Q1 or above Q3.

In this case1.5× IQR= 1.5× 3.5=5.25,Q1− 5.25=2.75, and Q3+ 5.25=16.75,so we can say that any data points below 2.75 or above 16.75 areoutliers. (Making a box-whisker plot is easier: see MATH200A Program part2.)

### Step 3: Find the variance.

Your TI-83 or TI-84 doesn’t find the variance for youautomatically, but since the standard deviation is the square root ofthe variance, you can**find the variance by squaring the standard deviation.**

It would be**wrong to compute s²= 3.17²**=10.05— see the Big No-no for thereason. You could enter 3.165257832², but that’s tedious anderror prone, as well as being overkill. Instead,

**use the value that the calculator has stored in a variable.**

Select statistics variables. | [`VARS` ] [`5` ] |

Select the correct standard deviation: `Sx` if your data set is a sample or `σx` if your data set is the whole population. | [`3` ] for `Sx` or [`4` ] for `σx` . |

Square it. The variance is s²=10.02. (If the data set was a whole population, you’d use σ² for the variance.) | [`x²` ] [`ENTER` ] |

## Descriptive Statistics for a Frequency Distribution

Class Boundaries | Class Marks | Frequency |
---|---|---|

20 ≤ x < 30 | 25 | 34 |

30 ≤ x < 40 | 35 | 58 |

40 ≤ x < 50 | 45 | 76 |

50 ≤ x < 60 | 55 | 187 |

60 ≤ x < 70 | 65 | 254 |

70 ≤ x < 80 | 75 | 241 |

80 ≤ x < 90 | 85 | 147 |

The grouped frequency distribution at right is the agesreported by Roman Catholic nuns, from [full citation at https://BrownMath.com/swt/sources.htm#so_Johnson2004], page 67.Let’s use the TI-83/84 to compute statistics.

### Step 1: Enter class marks in L1 and frequencies in L2.

By the way, this note uses L1 and L2, but you can use anylists you like, as long as you enter the actual list names in the`1-VarStats`

command in Step 2.(It doesn’t matter whether there are numbers in any other list.)

This example is for a grouped frequency distribution. If youhave an **ungrouped frequency distribution**, you can computestatistics in the same way. The only difference is that your firstlist will contain the actual values instead of the class marks.

Enter the class marks in `L1` . (The class mark is the midpoint of each class.) | [`STAT` ] [`1` ] selects the list-edit screen. Cursor onto the label `L1` at top of first column, then [`CLEAR` ] [`ENTER` ] erases the list. Enter the class marks. (If you have only the class boundaries, you can make the TI-83/84 do the work for you. It will compute the class marks automatically if you enter the class boundaries in the form `(20+30)÷2` .) |

Enter the frequencies in `L2` . | Cursor onto the label `L2` at top of first column, then [`CLEAR` ] [`ENTER` ] erases the list. Enter the frequencies. |

### Step 2: Compute the statistics.

Select the `1-VarStats` command. | [`STAT` ] [`►` ] [`1` ] pastes the command to the home screen. |

Specify which statistics lists contain the data set and the frequencies, in that order. Show your work: write down `1-VarStats` and both lists. Important: You must supply both lists. That’s the only way the calculator knows you have a frequency distribution. Always check the sample size n in the output, to guard against forgetting to enter the second list. If you see n is the number of classes instead of the number of data points, redo your `1-VarStats` and this time specify both lists. | Assuming you used `L1` and `L2` , enter [`2nd` `1` makes `L1` ] [`,` ] [`2nd` `2` makes `L2` ]. Press [ `ENTER` ] to execute the command. |

The important statistics are

**sample size***n*=997

Again, if this is a low number it means you forgot to specifyfrequencies on the`1-Var Stats`

command.**mean***x̅*=63.9

(Write symbol μ if this is a population mean.)**standard deviation***s*=15.4

If this data set is a sample, use`Sx`

and write*s*for thestandard deviation; if this data set is the whole population(including a probability distribution), use`σx`

and write σ for the standard deviation.**variance**is not shown on this screen; seeStep 3 below.

Remember that **the values on this screen are approximate**because the frequency distribution is an approximation of the originalraw data. For most real-life data sets, the approximation isquite good, and it is very good for moderate to large data sets.

The down arrow on the screen tells you that there’smore information if you scroll down. However, since the numbers youenter in a grouped frequency distribution are only approximate,the five-number summary is only approximate. The Min and Maxare just the highest and lowest classes. Q1, Med, and Q3 are at bestthe midpoints of the classes that actually contain thosestatistics.

As a general rule, **the five-number summary from a grouped frequency distribution is not worth reporting.**The numbers will be only approximations, becausethe calculator has only the class midpoints to work with and not theoriginal data.

### Step 3: Find the variance.

Just as with a simple list of numbers, you**find the variance by squaring the standard deviation.**

It would be**wrong to compute s²= 15.4²**=237.2— see the Big No-no for the reason.Instead, use the value that the calculator has stored in avariable.

Select statistics variables. | [`VARS` ] [`5` ] |

Select the correct standard deviation: `Sx` if your data set is a sample or `σx` if your data set is the whole population. | [`3` ] for `Sx` or [`4` ] for`σx` . |

Square it. The variance is s²=238.2. (If the data set was a whole population, the variance would be σ².) | [`x²` ] [`ENTER` ] |

## What’s New?

**20 Jan 2021**: In response to a reader’s query, clarifiedthe choice between`Sx`

and`σx`

, here and here.**9 Nov 2020**: Converted from HTML 4.01 to HTML5. Made a adozen or so small edits for clarity. Formatted math variable names andnames of variables seen on TI screens.- (Intervening changes suppressed.)
**6 Sep 2007**: New article.

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