# Sample Statistics on TI-83/84

Copyright © 2007–2021 by Stan Brown

Copyright © 2007–2021 by Stan Brown

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

**See also:**
MATH200A Program — Basic Statistics Utilities for TI-83/84 gives a downloadable program
to plot histograms and box-whisker diagrams.

**See also:**
optional advanced material: MATH200B Program part 1
gives a downloadable program that computes skewness and kurtosis, two
numerical measures of shape

Quiz scores in a (fictitious) class were 10.5, 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 sense of the class by just staring at the numbers, but you can easily compute the common measures of center and spread by using your TI-83 or TI-84.

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

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. |

Select the `1-Var Stats` 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 or entering 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 the standard 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; see Step 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. |
[`▼` 5 times ] for the five-number summary. |

You can tell the shape of the distribution. Since
the 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. Recall
that we use 1.5 × IQR to
classify outliers: we call a data point an outlier if it’s
at least that far below Q1 or above Q3.

In this case 1.5 × 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 are outliers. (Making a box-whisker plot is easier: see MATH200A Program part 2.)

Your TI-83 or TI-84 doesn’t find the variance for you
automatically, but since the standard deviation is the square root of
the 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 the
reason. You could enter 3.165257832², but that’s tedious and
error prone, as well as being overkill. Instead,

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` ] |

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 ages reported by Roman Catholic nuns, from Johnson & Kuby 2004 [full citation at https://BrownMath.com/swt/sources.htm#so_Johnson2004], page 67. Let’s use the TI-83/84 to compute statistics.

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

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 you
have an **ungrouped frequency distribution**, you can compute
statistics in the same way. The only difference is that your first
list 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. |

Select the `1-Var Stats` 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 specify frequencies 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 the standard 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; see Step 3 below.

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

The down arrow on the screen tells you that there’s more information if you scroll down. However, since the numbers you enter in a grouped frequency distribution are only approximate, the five-number summary is only approximate. The Min and Max are just the highest and lowest classes. Q1, Med, and Q3 are at best the midpoints of the classes that actually contain those statistics.

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

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

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

and`σx`

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

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