Descriptive Statistics: Five-Number Summary, Percentiles, Outliers, Box Plots, and Measures of Center

Finding the Median, Quartiles, and Five-Number Summary

• Procedure to Find the Median ($M$):

  • Sort the data values in ascending order from smallest to largest.

  • If the number of data values ($n$) is odd, the median is the single middle data value of the sorted data set.

  • If the number of data values ($n$) is even, the median is the average (arithmetic mean) of the two middle data values in the sorted data set.

• Procedure to Find the First Quartile ($Q_1$) and Third Quartile ($Q_3$):

  • For $Q_1$, isolate the lower half of the sorted data set (all values strictly below the median position). $Q_1$ is defined as the median of this lower half. The median itself must not be included in the lower half.

  • For $Q_3$, isolate the upper half of the sorted data set (all values strictly above the median position). $Q_3$ is defined as the median of this upper half. The median itself must not be included in the upper half.

• Five-Number Summary Components:

  • Minimum ($\text{Min}$)

  • First Quartile ($Q_1$)

  • Median ($\text{Median}$ or $M$)

  • Third Quartile ($Q_3$)

  • Maximum ($\text{Max}$)

• Example 1 ($n = 18$ Even Data Set):

  • Unsorted Data Set: $67, 94, 85, 79, 53, 78, 87, 98, 86, 67, 90, 77, 84, 98, 66, 92, 79, 89$

  • Sorted Data Set: $53, 66, 67, 67, 77, 78, 79, 79, 84, 85, 86, 87, 89, 90, 92, 94, 98, 98$

  • Total number of observations: $n = 18$

  • Median calculation: Average of the $9\text{th}$ value ($84$) and $10\text{th}$ value ($85$):     $\text{Median} = \frac{84 + 85}{2} = 84.5$

  • Lower half ($9$ data values): $53, 66, 67, 67, 77, 78, 79, 79, 84$

  • $Q_1 = 77$ (middle value of lower half)

  • Upper half ($9$ data values): $85, 86, 87, 89, 90, 92, 94, 98, 98$

  • $Q_3 = 90$ (middle value of upper half)

  • Five-Number Summary:

  • $\text{Min} = 53$

  • $Q_1 = 77$

  • $\text{Median} = 84.5$

  • $Q_3 = 90$

  • $\text{Max} = 98$

• Example 2 ($n = 11$ Odd Data Set):

  • Sorted Data Set: $51, 58, 62, 67, 72, 75, 78, 79, 85, 92, 99$

  • Total number of observations: $n = 11$

  • Median calculation: The middle ($6\text{th}$) data value is $75$.

  • Lower half ($5$ data values): $51, 58, 62, 67, 72$

  • $Q_1 = 62$

  • Upper half ($5$ data values): $78, 79, 85, 92, 99$

  • $Q_3 = 85$

Stem-and-Leaf Graphs

• Constructing a Stem-and-Leaf Plot for Sorting:

  • Constructing a stem-and-leaf plot is a convenient method for organizing and sorting raw data.

  • Example dataset: $67, 94, 85, 79, 53, 78, 87, 98, 86, 67, 90, 77, 84, 98, 66, 92, 79, 89$

  • Plot layout:

  • Stem $5$: Leaf $3$

  • Stem $6$: Leaves $6, 7, 7$

  • Stem $7$: Leaves $7, 8, 9, 9$

  • Stem $8$: Leaves $4, 5, 6, 7, 9$

  • Stem $9$: Leaves $0, 2, 4, 8, 8$

  • From the stem-and-leaf plot, key order statistics are readily read:

  • $\text{Min} = 53$

  • $Q_1 = 77$

  • $\text{Median} = 84.5$

  • $Q_3 = 90$

  • $\text{Max} = 98$

Identifying Outliers

• Definition of an Outlier:

  • An outlier is an observation or data value that does not fit the rest of the data because it is unusually low or unusually high.

• Interquartile Range (IQR) Formula:   $\text{IQR} = Q_3 - Q_1$

• Outlier Boundary Formulas (Fences):

  • A data value is defined as a suspected potential outlier if it falls beyond either of the following boundaries:

  • Less than $1.5 \times \text{IQR}$ below the first quartile ($Q_1$):       $\text{Left Fence} = Q_1 - 1.5 \times (\text{IQR})$

  • More than $1.5 \times \text{IQR}$ above the third quartile ($Q_3$):       $\text{Right Fence} = Q_3 + 1.5 \times (\text{IQR})$

• Example Outlier Calculation:

  • Given dataset: $53, 66, 67, 67, 77, 78, 79, 79, 84, 85, 86, 87, 89, 90, 92, 94, 98, 98$

  • $Q_1 = 77$

  • $Q_3 = 90$

  • Calculate Interquartile Range:     $\text{IQR} = 90 - 77 = 13$

  • Calculate Left Fence:     $\text{Left Fence} = Q_1 - 1.5 \times (\text{IQR}) = 77 - 1.5 \times (13) = 77 - 19.5 = 54.5$

  • Calculate Right Fence:     $\text{Right Fence} = Q_3 + 1.5 \times (\text{IQR}) = 90 + 1.5 \times (13) = 90 + 19.5 = 109.5$

  • Outlier Determination: The data value $53$ is less than the left fence of $54.5$ (53 < 54.5), so $53$ is suspected to be a potential outlier.

Calculating Percentiles and Quartiles from Frequency Tables

• Frequency Table of Student Ages ($n = 50$):      | Age | Frequency | Relative Frequency | Cumulative Relative Frequency |   | :--- | :--- | :--- | :--- |   | $17$ | $2$ | $0.04$ | $0.04$ ($4\%$) |   | $18$ | $4$ | $0.08$ | $0.12$ ($12\%$) |   | $19$ | $7$ | $0.14$ | $0.26$ ($26\%$) |   | $20$ | $8$ | $0.16$ | $0.42$ ($42\%$) |   | $21$ | $6$ | $0.12$ | $0.54$ ($54\%$) |   | $22$ | $14$ | $0.28$ | $0.82$ ($82\%$) |   | $23$ | $9$ | $0.18$ | $1.00$ ($100\%$) |   | Total | $50$ | $1.00$ | $1.00$ |

• Determining Percentiles from Cumulative Relative Frequencies:

  • $40\text{th}$ Percentile: Look for the age value where cumulative relative frequency reaches or first encompasses $0.40$ ($40\%$). Cumulative relative frequency reaches $0.26$ at age $19$ and $0.42$ at age $20$. Therefore, the $40\text{th}$ percentile is $20$.

  • Third Quartile ($Q_3$ / $75\text{th}$ Percentile): Look for the age value where cumulative relative frequency reaches or exceeds $0.75$ ($75\%$). Cumulative relative frequency reaches $0.54$ at age $21$ and $0.82$ at age $22$. Therefore, the third quartile is $22$.

  • Median ($50\text{th}$ Percentile): Look for the age value where cumulative relative frequency reaches or exceeds $0.50$ ($50\%$). Cumulative relative frequency reaches $0.42$ at age $20$ and $0.54$ at age $21$. Therefore, the median is $21$.

  • $12\text{th}$ Percentile: At age $18$, the cumulative relative frequency is exactly $0.12$ ($12\%$), representing the $6\text{th}$ observation out of $50$ ($2 + 4 = 6$, and $\frac{6}{50} = 0.12$). Since $0.12$ lands exactly on the boundary between age $18$ and age $19$, average the $6\text{th}$ data value ($18$) and $7\text{th}$ data value ($19$):     $\text{12th Percentile} = \frac{18 + 19}{2} = 18.5$

Calculating the kth Percentile Using a Formula

• Formula Definitions:

  • Let $k$ be the desired percentile number.

  • Let $n$ be the total number of data values in the data set.

• Step-by-Step Algorithm:

  1. Sort the data set from smallest to largest.

  2. Compute the index $i$ using the formula:      $i = \frac{k}{100} \times (n + 1)$

  3. Determine the percentile value based on $i$:

  • Case A: If $i$ is an integer, the $i\text{th}$ data value in the sorted set is the $k\text{th}$ percentile.

  • Case B: If $i$ is not an integer, round $i$ down to the nearest lower integer $j$, and round $i$ up to the next integer $j + 1$. The $k\text{th}$ percentile is the average of the $j\text{th}$ and $(j+1)\text{th}$ data values.

• Example ($k = 62$, $n = 30$):

  • Sorted Data Set ($30$ values):     $32.5, 34, 37, 41.5, 41.5, 45, 47.5, 50.5, 50.5, 50.5, 53, 58.5, 61, 64, 66, 67.5, 68, 68, 72, 75, 75.5, 77, 79, 79, 84.5, 89, 92.5, 96, 102.5, 114$

  • Calculate index $i$:     $i = \frac{62}{100} \times (30 + 1) = 0.62 \times 31 = 19.22$

  • Since $i = 19.22$ is not an integer, round down to $j = 19$ and up to $j + 1 = 20$

  • Locate the $19\text{th}$ and $20\text{th}$ values:

  • $19\text{th} \text{ value} = 72$

  • $20\text{th} \text{ value} = 75$

  • Calculate the $62\text{nd}$ percentile:     $\text{62nd Percentile} = \frac{72 + 75}{2} = 73.5$

• Mathematical Justification for $i = \frac{k}{100}(n+1)$ vs. $i = \frac{k}{100}n$:

  • Consider finding the $40\text{th}$ percentile ($k = 40$) for dataset ($n = 10$): $1, 3, 5, 7, 9, 11, 13, 15, 17, 19$

  • If using $i = \frac{k}{100}n = \frac{40}{100} \times 10 = 4$:

  • The $4\text{th}$ value is $7$. However, only $3$ values ($1, 3, 5$) are strictly less than $7$. Since $\frac{3}{10} = 30\%$, $7$ represents the $30\text{th}$ percentile, not the $40\text{th}$ percentile.

  • Correct approach using $i = \frac{k}{100}(n+1) = \frac{40}{100} \times (10 + 1) = \frac{40}{100} \times 11 = 4.4$:

  • Take the average of the $4\text{th}$ value ($7$) and $5\text{th}$ value ($9$):       $\text{40th Percentile} = \frac{7 + 9}{2} = 8$

Finding the Percentile Rank of a Given Value

• Variable Definitions:

  • Let $x =$ number of data values counting from the bottom of the data list up to, but not including, the data value for which you want to find the percentile.

  • Let $y =$ number of data values equal to the data value for which you want to find the percentile (including the data value itself).

  • Let $n =$ total number of data values in the data set.

• Procedure:

  1. Sort the data set from smallest to largest.

  2. Calculate the raw percentile value using:      $\text{Percentile} = \frac{x + 0.5y}{n} \times 100$

  3. Round off this final calculated value to the nearest integer.

• Example (Finding the Percentile Rank of $50.5$ in $n = 30$ Dataset):

  • Sorted Dataset:     $32.5, 34, 37, 41.5, 41.5, 45, 47.5, 50.5, 50.5, 50.5, 53, 58.5, 61, 64, 66, 67.5, 68, 68, 72, 75, 75.5, 77, 79, 79, 84.5, 89, 92.5, 96, 102.5, 114$

  • Count values strictly less than $50.5$: $x = 7$ (values: $32.5, 34, 37, 41.5, 41.5, 45, 47.5$)

  • Count values equal to $50.5$: $y = 3$ (values: $50.5, 50.5, 50.5$)

  • Total number of observations: $n = 30$

  • Calculation:     $\text{Percentile} = \frac{7 + 0.5(3)}{30} \times 100 = \frac{7 + 1.5}{30} \times 100 = \frac{8.5}{30} \times 100 = 28.3333...$

  • Rounding to the nearest integer gives $28\text{th}$ percentile.

Box Plots (Box-and-Whisker Plots)

• Structure and Construction of a Box Plot:

  • The Box:

  • Left edge is drawn at $Q_1$

  • Right edge is drawn at $Q_3$

  • Vertical line inside the box is drawn at the Median ($\text{Med}$ or $M$)

  • Length of the box equals the Interquartile Range ($\text{IQR} = Q_3 - Q_1$)

  • The Whiskers:

  • Line segments extending outwards from the box

  • Left whisker extends from $Q_1$ down to the Minimum ($\text{Min}$)

  • Right whisker extends from $Q_3$ up to the Maximum ($\text{Max}$)

  • Small vertical tick marks are drawn at the outer terminal ends of both whiskers

  • Total Length:

  • Length of the entire plot equals the Range ($\text{Range} = \text{Max} - \text{Min}$)

• Distribution Percentages within a Box Plot:

  • Each section between summary marks represents exactly $25\%$ of the data:

  • $\text{Min}$ to $Q_1$: $25\%$ of data values

  • $Q_1$ to Median: $25\%$ of data values

  • Median to $Q_3$: $25\%$ of data values

  • $Q_3$ to $\text{Max}$: $25\%$ of data values

  • The middle $50\%$ of all data values fall inside the central box ($25\% + 25\% = 50\%$).

• Example 1:

  • Summary: $\text{Min} = 10$, $Q_1 = 50$, $\text{Median} = 70$, $Q_3 = 80$, $\text{Max} = 85$

  • Box extends from $50$ to $80$ (length = $30$).

  • Median mark at $70$.

  • Left whisker extends from $50$ down to $10$.

  • Right whisker extends from $80$ up to $85$.

• Example 2:

  • Summary: $\text{Min} = 53$, $Q_1 = 77$, $\text{Median} = 84.5$, $Q_3 = 90$, $\text{Max} = 98$

  • Box extends from $77$ to $90$ (length = $13$).

  • Median mark at $84.5$.

  • Left whisker extends from $77$ down to $53$.

  • Right whisker extends from $90$ up to $98$.

Graphing Box Plots on a TI Calculator

• Step-by-Step Calculator Steps:

  1. Press STAT, select 1:Edit. Clear existing list data if present (arrow up into heading L1, press CLEAR, then ENTER or press STAT, 4:ClrList, enter 2nd 1 for L1, press ENTER).

  2. Input all raw numerical data values into list L1.

  3. Press STAT, arrow right to CALC, select 1:1-Var Stats. Verify list is set to L1, press ENTER. Scroll using arrow keys to view five-number summary ($\text{minX}$, $Q_1$, $\text{Med}$, $Q_3$, $\text{maxX}$).

  4. Press 2nd Y= (STAT PLOT). Select 4:PlotsOff, press ENTER to turn off prior plots.

  5. Press 2nd Y=, select 1:Plot1, press ENTER. Select On. Arrow down to Type, arrow right to choose the box plot icon ($5\text{th}$ icon picture option), press ENTER.

  6. Set Xlist to L1 (2nd 1). Set Freq to 1.

  7. Press ZOOM, select 9:ZoomStat.

  8. Press TRACE and use left/right arrow keys to display the values of $\text{Min}$, $Q_1$, $\text{Med}$, $Q_3$, and $\text{Max}$ on the box plot.

Measures of Center and Summation Notation

• Primary Measures of Central Tendency:

  • Median ($M$): The physical middle value of an ordered dataset.

  • Mean (Average): The arithmetic balance point calculated by summing values and dividing by count.

  • Mode: The value that appears most frequently in the dataset.

• Summation Notation (Sigma Notation):

  • Let $x_1, x_2, x_3, \dots, x_n$ represent $n$ individual observations.

  • The summation symbol $\sum x_i$ denotes the total sum of all observations:     $\sum x_i = x_1 + x_2 + x_3 + \dots + x_n$

• Example (Top Four Grossing Superhero Movies):      | Movie Title | Gross ($x_i$ in millions of dollars) |   | :--- | :--- |   | Black Panther | $700$ ($x_1$) |   | Avengers: Infinity War | $679$ ($x_2$) |   | Marvel's The Avengers | $623$ ($x_3$) |   | Incredibles 2 | $609$ ($x_4$) |

  • Number of observations: $n = 4$

  • Individual values: $x_1 = 700$, $x_2 = 679$, $x_3 = 623$, $x_4 = 609$

  • Sum of observations ($\sum x_i$):     $\sum x_i = x_1 + x_2 + x_3 + x_4 = 700 + 679 + 623 + 609 = 2611$

  • Contextual meaning: The total combined box office gross of these four movies is $\$2,611,000,000$ ($2.611$ billion dollars).

  • Sum of squared observations ($\sum x_i^2$):     $\sum x_i^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 = 700^2 + 679^2 + 623^2 + 609^2$     $\sum x_i^2 = 490000 + 461041 + 388129 + 370881 = 1710051$

Sample Mean vs. Population Mean

• General Definition of Mean:

  • For a data set with $n$ observations $x_1, x_2, x_3, \dots, x_n$, the mean is:     $\text{Mean} = \frac{\sum x_i}{n}$

  • Calculation steps:

  1. Add together all individual data values ($x_1$ through $x_n$).

  2. Divide this computed total sum by the total count of values ($n$).

• Sample Mean (Statistic):

  • Calculated from a sample of $n$ observations ($x_1, x_2, \dots, x_n$).

  • Denoted by $\bar{x}$ (read as "x-bar"):     $\bar{x} = \frac{\sum x_i}{n}$

• Population Mean (Parameter):

  • Calculated from all $N$ observations in an entire population ($x_1, x_2, \dots, x_N$).

  • Denoted by $\mu$ (Greek letter "mu"):     $\mu = \frac{\sum x_i}{N}$

• Example (Parking Ticket Fines):

  • Data set of fines (in dollars) for $12$ randomly selected parking tickets in Los Angeles on March 9, 2019:     $93, 93, 73, 68, 93, 68, 68, 68, 73, 73, 68, 73$

  • Sorted Data Set ($n = 12$):     $68, 68, 68, 68, 68, 73, 73, 73, 73, 93, 93, 93$

  • Calculate Sample Mean ($\bar{x}$):     $\sum x_i = 68 + 68 + 68 + 68 + 68 + 73 + 73 + 73 + 73 + 93 + 93 + 93 = 911$     $\bar{x} = \frac{911}{12} \approx 75.9167$

  • Calculate Sample Median ($M$):

  • Average of $6\text{th}$ value ($73$) and $7\text{th}$ value ($73$):       $\text{Sample Median} = \frac{73 + 73}{2} = 73$

TI Calculator Instructions for Mean and Median

• Step-by-Step Procedure:

  1. Clear list L1: Press STAT, select 4:ClrList, enter 2nd 1 (for L1), and press ENTER.

  2. Input raw data values into the list editor: Press STAT, select 1:Edit, and enter each data value into L1.

  3. Calculate summary statistics: Press STAT, arrow right to CALC, select 1:1-Var Stats, enter L1 (2nd 1), and press ENTER.

  4. Read outputs: Scroll through the output list using the arrow keys. The sample mean is displayed as $\bar{x}$, and the median is displayed as $M$ or $\text{Med}$.