Data lists and frequency distributions

This looks at data presented in frequency tables with precise values (not intervals). We view enumerated (listed out) data as a special case where the frequency is $1$ .

Major re-write: 2026-03-24.

Frequency table data
Formulas
- mean
- standard deviation and variance
Technology
Median
Quartiles
Interquartile range and outliers
Box and whisker diagram

Frequency table data

Example: An IBDP Year 1 math class has $17$ students. Here is their grade distribution.

Grade, $x_i$	$2$	$3$	$4$	$5$	$6$	$7$
Frequency, $f_i$	$3$	$4$	$6$	$3$	$0$	$1$

Formulas

The formulas are not necessarily there for you to compute values by hand in Paper 1, but they may allow you to set up an equation solve for an unknown value with help of calculators in Paper 2. The purpose of the examples are to illustrate the formulas.

mean

\bar{x} = \frac1n\sum_{i=1}^n f_i x_i

For our example

\begin{align*} \bar{x} &= \frac1{17}(3\cdot2+4\cdot3 + 6\cdot4 + 3\cdot5 + 0\cdot6 + 1\cdot7) \\ &= \frac{64}{17} = 3.7647\dots \qed \end{align*}

standard deviation and variance

SL students should be able to compute standard deviation and variance when given a GDC.

HL students need to know the following formulas.

Formula for variance：

\sigma^2 = \frac1n\sum_{i=1}^n f_i (x_i - \bar x)^2

but usually we use

\sigma^2 = \frac1n\sum_{i=1}^n f_i x_i^2 - \left(\bar{x}\right)^2

Note that the $-\left(\bar x\right)^2$ is outside the summation.

A proof for the special case of $f_i = 1$ is provided at Sigma notation, sequences, series, finance..

For our example

\begin{align*} \sigma^2 &= \frac1{17}\left(3\cdot2^2+4\cdot3^2 + 6\cdot4^2 + 3\cdot5^2 + 0\cdot6^2 + 1\cdot7^2\right) - \left(\frac{64}{17}\right)^2 \\ &= \frac{268}{17} - \left(\frac{64}{17}\right)^2 \\ &= \frac{460}{289} \qed \end{align*}

The standard deviation, $\sigma$ in our case would be

\sigma = \sqrt{\frac{460}{289}} = \frac{\sqrt{460}}{17} = 1.261624\dots \qed

Technology

In general we use lists and 1-variable statistics on our GDC to compute statistics. Provide a list of values $x$ followed by a list of $f$ frequencies.

TI 84 Plus CE TI Nspire CX Casio fx-CG50 HP Prime NumWorks

In stat 1:Edit, enter the values in L1 and the frequencies in L2.

store values in L1 and L2 — data entry on TI 84 Plus CE

In stat CALC, choose 1:1-Var Stats. L1 can be retrieved from 2nd 1.

1-Var Stats. List:L1. FreqList:L2. Calculate — 1 var stats settings

1-Var Stats. Showing values for mean, sum of x, sum of x squared, Sx, σx, n, minX, Q1 — first part of 1 var stats results

1-Var Stats. Showing Q1, Median, Q3, maxX — second part of 1 var stats results

To retrieve statistics such as σx, use vars 5:Statistics.

main screen. σx. 1.261624152 — standard deviation

Do not use 2nd stat Math 7:stdDev(, as that is for Sx.

Using List & Spreadsheets App, store $x$ list in A and $f$ list in B.

In menu 4:Statistics > 1:Stat Calculations > 1:One-Variable Statistics, set Num of lists to 1.

store values in a and b, then set num of lists to 1 — set num of lists to 1

Set X1 list, Frequency list, 1st Result Column. Use square brackets from ctrl (

Results showing mean x, sum x, sum x squared, sx — the results are shown and saved in variable "Stat."

Results showing sx, σx, n, MinX, Q1X — standard deviation is 1.2616

Results showing Q1X, MedianX, Q3X, MaxX, SSX — rest of results

To use any of these values eg standard deviation, use var and find stat.σx.

Tip: You can also just ctrl C, ctrl V. I like to ctrl var the copied value in a variable before using.

In Statistics (STAT) App, enter $x$ values in List 1, and $f$ values in List 2.

store values List 1, and frequencies in List 2

In CALC, SET add the suitable lists. To enter list 1, press LIST and enter 1.

Exit and use 1-VAR to see the statistics.

1-Variable Results for mean of x, sum of x, sum of x squared, σx, sx, n. — 1-variable statistics, first part

1-Variable results for n, minX, Q1, median, Q3, maxX — rest of results showing quartiles

To use one of the values, in VARS, STAT, X and choose for instance σx.

In Statistics 1Var app, enter the $x$ values in D1 and $f$ values in D2.

Statistics 1Var Numeric View. Values in D1. Frequencies in D2 — data entry in D1 and D2

In Symb ✗ , H1, enter D1 and D2 together in that row. They are available from Column button on screen.

Statistics 1Var Symbolic View. H1 checked. H1: D1 D2 — Enter both D1 and D2 when asked for H1 in Symbolic View

Go back to Num ⊞ , press Stats to view the statistics.

Statistics 1Var Numeric View. Results showing n, quartiles, sum of x, sum of x squared, mean of x, sx. — Statistics 1Var Numeric View, showing the statistics

Statistics 1Var Numeric View. Results quartiles, sum of x, sum of x squared, mean of x, sx, σx, serrX, ssX — Statistics 1Var Numeric View, showing the last part

Press OK to go back. To use a particular statistics, you can use Vars App Statistics 1Var Results and choose for example σx.

Statistics 1Var. Menus showing how to access σx — Retrieving σx for use in a calculation

Tip: You can also just Shift ⧉View , Shift Menu . I like to ▶ the copied value in a variable before using.

In Statistics app, enter values in V1 and frequencies in N1.

Data tab highlights, shows values in column V1 and frequenceis in N1 — entering the frequency table in NumWorks

Use arrow keys to the Stats tab to see the statistics.

Stats tab. Showing n, quartiles, range, IQR. — first half of the statistics

Stats tab. Showing mean, population standard deviation and other statistics — second half of the statistics

To use a statistic, copy-paste using shift var, shift ⊟. I like to shift xʸ the copied value in a variable before using.

In addition to mean x̄, standard deviation σx, and quartiles, notice the relevance of $64$ as Σx, and $268$ as Σx² in the above calculations.

Median

Sort the list.

For an odd $n$ number of data, the median is the value at the $\displaystyle \frac{n+1}{2}$ th position.

For an even $n$ number of data, the median is the mean between the $\displaystyle \frac{n}{2}$ th and $\displaystyle \frac{n}{2}+1$ st position.

Calculating medians by hand for frequency tables is required for SL and HL.

Quartiles

Q1, or 25th percentile, is greater than 25% of the data.

Median, or Q2, or 50th percentile, is greater than 50% of the data.

Q3, or 75th percentile, is greater than 75% of the data.

Q1 and Q3 should be found using 1-variable statistics on the calculator, as shown above.

Interquartile range and outliers

\text{IQR} = Q3 - Q1

Outliers lies more than $1.5 \times$ the interquartile range away from the nearest quartile.

Outliers don’t necessarily need to be thrown away. In contrast, they can be central in many scientific studies. They could mean measuring a value that has a very small chance of occurring, observing an effect that only impacts extreme conditions, in addition to indicating a potential mistake.

Presence of an outlier does not require recomputing any statistical value or measure, unless otherwise specified in the question.

For our example, $\text{IQR} = Q3-Q1=1.5$ . We want to check if $7$ is an outlier. It is closest to $Q3=4.5$ .

4.5 + 1.5(1.5) = 6.75 \leq 7

Therefore, $7$ is an outlier.

Box and whisker diagram

The boxes are drawn from Q1 to Q2, and Q2 to Q3. Lines extend from the minimum to Q1, and from Q3 to the max. Outliers are shown as a cross.

Here in our example, the box and whisker diagram has a box from $Q1$ to $Q2$ and from $Q2$ to $Q3$ . There are lines (whiskers) from minimum to $Q1$ and from $Q3$ to maximum. There is an outlier at $7$ marked with a cross.

Outliers only affect the min and max values drawn, and not the boxes.

Mode and mean cannot be determined from a box and whisker plot; instead they require access to the original, individual values.

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Next : Grouped data

Data lists and frequency distributions

Contents

Frequency table data

Formulas

mean

standard deviation and variance

Technology

Median

Quartiles

Interquartile range and outliers

Box and whisker diagram