Cut every value into a stem and a leaf, read counts and far values off the finished plot, put two data sets on one stem column, join counted points across equal steps, and turn head counts into shares of a named whole.
Subject: Statistics · 65 slides · applied lesson
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Title
Statistics · §2.1
Four pictures for data, each drawn from the numbers it shows
Objectives
Concept
\[ 23 = 2 \times 10 + 3 \]
Place value
Why: tens and ones name a number
\[ 5,\ 1,\ 3 \to 1,\ 3,\ 5 \]
Sort
Why: smallest value written first
\[ u + v = v + u \]
Any order
Why: a total ignores the arrangement
\[ u \div v = w \iff w \times v = u \]
Divide back
Why: checks every share
\[ 0.25 = 25\% \]
Per cent
Why: % counts parts per hundred
\[ u - v \]
Subtract
Why: measures the gap between two values
\[ 1,\ 2,\ 3,\ 4 \text{ evenly spaced} \]
Equal steps
Why: matching gaps mean matching amounts
\[ 2 \div 3 \approx 0.67 \]
Round
Why: ≈ marks a rounded value
Prediction
Figure (svg): Example 2.1's 31 exam scores in cells, smallest first, from 33 to 100
Predict first
Professor Dean's 31 first-exam scores are written out, smallest to largest.
Which of these can you read off without counting?
Correct: The lowest score
Why: A sorted list hands you its two ends for free: the first cell is the lowest score and the last is the highest. Every other question here makes you walk the whole list and hold a running tally in your head.
Worked example
Figure (svg): Example 2.1's 31 exam scores in cells, smallest first, from 33 to 100
\[ \textcolor{#1f5fbf}{61},\ \textcolor{#1f5fbf}{63},\ \textcolor{#1f5fbf}{67},\ \textcolor{#1f5fbf}{68},\ \textcolor{#1f5fbf}{68},\ \textcolor{#1f5fbf}{69},\ \textcolor{#1f5fbf}{69} \]
Pick out the 60s scores
Why: a tally needs them separated first
Figure (svg): Example 2.1's 31 exam scores in cells, smallest first, from 33 to 100; 7 cells highlighted
\[ \text{how many} = 7 \]
Count the picked cells
Why: the sixth ten's height in a picture
\[ 31 - 7 = 24 \]
Subtract 7 from 31
Why: cells read for nothing
\[ \textcolor{#1f5fbf}{69},\ \textcolor{#1f5fbf}{69},\ \textcolor{#1f5fbf}{68},\ \textcolor{#1f5fbf}{68},\ \textcolor{#1f5fbf}{67},\ \textcolor{#1f5fbf}{63},\ \textcolor{#1f5fbf}{61} \]
Check: count the 60s backwards
Why: the same seven cells in reverse
Needed: one picture that counts and keeps scores.
Section
Idea 1 of 4
Concept
Figure (svg): Example 2.1's 31 exam scores in cells, smallest first, from 33 to 100
Goal: one picture holding counts and scores.
Discussion prompt
What would you lose by replacing each score with the ten it falls in?
Answer:
The score itself: 61 and 69 would both become '60s'.
Worked example
Figure (svg): A bar for each ten of Example 2.1's scores, drawn to scale on a count axis from 0 to 8: 30s 1, 40s 3, 50s 3, 60s 7, 70s 4, 80s 5, 90s 7, 100s 1
\[ \textcolor{#1f5fbf}{33} \to \text{the 30s} \]
Bin 33 with the 30s
Why: each bin covers one ten
Figure (svg): A bar for each ten of Example 2.1's scores, drawn to scale on a count axis from 0 to 8: 30s 1, 40s 3, 50s 3, 60s 7, 70s 4, 80s 5, 90s 7, 100s 1
\[ 1,\ 3,\ 3,\ 7,\ 4,\ 5,\ 7,\ 1 \]
Tally the eight tens
Why: bar lengths come from counts
Figure (svg): A bar for each ten of Example 2.1's scores, drawn to scale on a count axis from 0 to 8: 30s 1, 40s 3, 50s 3, 60s 7, 70s 4, 80s 5, 90s 7, 100s 1
\[ 1 + 3 + 3 + 7 + 4 + 5 + 7 + 1 = 31 \]
Add the eight counts
Why: no score sits outside a bin
\[ \text{the 30s bar} = 1 \]
Read the 30s bar
Why: tests whether a score returns
Figure (svg): A bar for each ten of Example 2.1's scores, drawn to scale on a count axis from 0 to 8: 30s 1, 40s 3, 50s 3, 60s 7, 70s 4, 80s 5, 90s 7, 100s 1; the 30s bar thickened
\[ 30 \le \text{that score} \le 39 \]
Check: bound the hidden score
Why: ten values could draw it
Worked example
Figure (svg): Three of Professor Dean's scores in cells: 68, 69 and 72
\[ \textcolor{#1f5fbf}{68} = 6 \times 10 + \textcolor{#1f5fbf}{8} \]
Split 68 into tens and ones
Why: place value offers two parts
Figure (svg): Values cut into a boxed stem and a boxed leaf, with a dashed line marking the cut
\[ \text{stem } 6, \quad \text{leaf } \textcolor{#1f5fbf}{8} \]
Name the parts stem and leaf
Why: filing a score needs its tens
\[ \textcolor{#1f5fbf}{72} = 7 \times 10 + \textcolor{#1f5fbf}{2} \]
Cut 72 the same way
Why: its tens digit opens a row
Figure (svg): A stemplot with rows 6, 7 and 8 and no leaves yet
\[ 6 \mid \textcolor{#1f5fbf}{8}\ \textcolor{#1f5fbf}{9}, \quad 7 \mid \textcolor{#1f5fbf}{2} \]
File each leaf beside its stem
Why: one row per ten of scores
Figure (svg): The stemplot with 8 and 9 on row 6 and 2 on row 7
\[ 6 \times 10 + \textcolor{#1f5fbf}{9} = \textcolor{#1f5fbf}{69} \]
Check: rebuild 69 from its row
Why: the leaf digit survived the filing
Worked example
Figure (svg): Values cut into a boxed stem and a boxed leaf, with a dashed line marking the cut
\[ \textcolor{#1f5fbf}{432} = 43 \times 10 + \textcolor{#1f5fbf}{2} \]
Write 432 as tens and ones
Why: the cut ignores a number's size
Figure (svg): Values cut into a boxed stem and a boxed leaf, with a dashed line marking the cut
\[ \textcolor{#1f5fbf}{5432} = 543 \times 10 + \textcolor{#1f5fbf}{2} \]
Write 5432 as tens and ones
Why: a stem takes any digits left
Figure (svg): Values cut into a boxed stem and a boxed leaf, with a dashed line marking the cut
\[ \textcolor{#1f5fbf}{9.3}: \text{final digit} = \textcolor{#1f5fbf}{3} \]
Cut 9.3 after the point
Why: decimals earn leaves too
Figure (svg): Values cut into a boxed stem and a boxed leaf, with a dashed line marking the cut
\[ \textcolor{#1f5fbf}{\text{leaf}} = \text{final digit}, \ \ \text{stem} = \text{the rest} \]
State the rule for any value
Why: one leaf digit fixes row width
\[ 543 \mid \textcolor{#1f5fbf}{2} \to \textcolor{#1f5fbf}{5432} \]
Check: read 5432 back out
Why: no digit was thrown away
Worked example
Figure (svg): The stemplot with 8 and 9 on row 6, 2 on row 7, and row 8 blank
\[ 6 \mid \textcolor{#1f5fbf}{8}\ \textcolor{#1f5fbf}{9} \to 2 \text{ leaves} \]
Count row 6's leaves
Why: row length is that ten's height
\[ 8 \mid \quad \to 0 \text{ scores} \]
Read the blank row 8
Why: an empty ten still owns a row
Figure (svg): The stemplot with rows 6 and 7 filled and the blank row 8 shaded
\[ 6 \times 10 = 60 \]
Multiply row 6's stem by ten
Why: the stem stands for whole tens
\[ 60 + \textcolor{#1f5fbf}{8} = \textcolor{#1f5fbf}{68} \]
Add the first leaf
Why: the leaf restores the ones digit
\[ \textcolor{#1f5fbf}{68} - \textcolor{#1f5fbf}{8} = 60 \]
Check: take the leaf back off
Why: the stem's sixty is what remains
Worked example
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 0 rows
\[ \textcolor{#1f5fbf}{33} \to 3 \mid \textcolor{#1f5fbf}{3} \]
File the lowest score
Why: row 3 opens the plot
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 1 rows
\[ \textcolor{#1f5fbf}{42},\ \textcolor{#1f5fbf}{49},\ \textcolor{#1f5fbf}{49} \to 4 \mid \textcolor{#1f5fbf}{2}\ \textcolor{#1f5fbf}{9}\ \textcolor{#1f5fbf}{9} \]
File the three 40s scores
Why: list order is kept
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 2 rows
\[ \textcolor{#1f5fbf}{53},\ \textcolor{#1f5fbf}{55},\ \textcolor{#1f5fbf}{55} \to 5 \mid \textcolor{#1f5fbf}{3}\ \textcolor{#1f5fbf}{5}\ \textcolor{#1f5fbf}{5} \]
File the three 50s scores
Why: a tie still earns two leaves
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 3 rows
\[ \textcolor{#1f5fbf}{61} \ldots \textcolor{#1f5fbf}{69} \to 6 \mid \textcolor{#1f5fbf}{1}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{9}\,\textcolor{#1f5fbf}{9} \]
File the seven 60s scores
Why: every score in the sixties belongs here
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 4 rows
\[ 1 + 3 + 3 + 7 = 14 \]
Check: add the four rows
Why: 14 scores end at 69
Worked example
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 4 rows
\( {3 \mid 3},\ \allowbreak \allowbreak {4 \mid 2\,9\,9},\ \allowbreak \allowbreak {5 \mid 3\,5\,5},\ \allowbreak \allowbreak {6 \mid 1\,3\,7\,8\,8\,9\,9} \)
\[ \textcolor{#1f5fbf}{72},\ \textcolor{#1f5fbf}{73},\ \textcolor{#1f5fbf}{74},\ \textcolor{#1f5fbf}{78} \to 7 \mid \textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{8} \]
File the four 70s scores
Why: one row holds one ten
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 5 rows
\[ \textcolor{#1f5fbf}{80} \ldots \textcolor{#1f5fbf}{88} \to 8 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{8} \]
File the five 80s scores
Why: repeats keep their own leaves
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves filled for 6 rows
\[ 9 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6}; \ \ 10 \mid \textcolor{#1f5fbf}{0} \]
File the 90s and 100
Why: nothing left over
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves
\[ 1 + 3 + 3 + 7 + 4 + 5 + 7 + 1 = 31 \]
Check: add every row
Why: 31 students, none lost
Worked example
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves
\[ 9 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6} \to 7 \]
Count row 9's leaves
Why: the high grades begin here
Figure (svg): Example 2.1's stemplot with row 9 shaded
\[ 10 \mid \textcolor{#1f5fbf}{0} \to 1 \]
Count row 10's leaf
Why: 100 joins the high grades
Figure (svg): Example 2.1's stemplot with rows 9 and 10 shaded
\[ 7 + 1 = 8 \]
Add the two row lengths
Why: how many earned 90 or above
\[ 8 \div 31 \approx 0.258 \]
Divide 8 by 31
Why: a share needs the whole class
\[ 0.258 \to \textcolor{#b54708}{25.8\%} \]
Read 0.258 as a per cent
Why: % rescales it to a hundred
\[ 0.258 \times 31 \approx 8 \]
Check: undo the division
Why: the share returns eight students
Trap
Figure (svg): A stemplot of the Hawks' losses in their first ten seasons (Table 2.6, seasons 1 to 10) with the leaves in season order: row 3 reads 4 4 6, row 4 reads 6 6 7 1, row 5 reads 1 3 1
\[ \textcolor{#1f5fbf}{34},\ \textcolor{#1f5fbf}{34},\ \textcolor{#1f5fbf}{36} \to 3 \mid \textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6} \]
Copy the 30s losses unsorted
Why: season order reaches the plot
\[ \textcolor{#1f5fbf}{46},\ \textcolor{#1f5fbf}{46},\ \textcolor{#1f5fbf}{47},\ \textcolor{#1f5fbf}{41} \to 4 \mid \textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{1} \]
Copy the 40s losses unsorted
Why: fails: 41 ends up last
Figure (svg): The same stemplot with every row sorted: row 3 reads 4 4 6, row 4 reads 1 6 6 7, row 5 reads 1 1 3
\[ \textcolor{#1f5fbf}{46},\ \textcolor{#1f5fbf}{46},\ \textcolor{#1f5fbf}{47},\ \textcolor{#1f5fbf}{41} \to 4 \mid \textcolor{#1f5fbf}{1}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{7} \]
Sort the 40s losses first
Why: a row must read low to high
\[ 4 \mid \textcolor{#1f5fbf}{1} \to \textcolor{#1f5fbf}{41} \]
Check: read the row's first leaf
Why: 41 sits under every other leaf
Prediction
Figure (svg): A bar for each ten of Example 2.1's scores, drawn to scale on a count axis from 0 to 8: 30s 1, 40s 3, 50s 3, 60s 7, 70s 4, 80s 5, 90s 7, 100s 1
Predict first
One picture bins the scores by tens; the other files them by stem and leaf.
Which question can only the stemplot answer?
Correct: What was the lowest score?
Why: A bar reports a count, so any question about counts can be read off either picture. Only the stemplot still carries the digits, so only it can say the lowest score was 33 rather than 'somewhere from 30 to 39'.
Worked example
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves
\[ \text{the 30s bar} = 1 \]
Read the first bar
Why: a bar gives only a height
\[ 30 \le \text{that score} \le 39 \]
Bound the score from its bin
Why: ten values fit there
\[ 3 \mid \textcolor{#1f5fbf}{3} \]
Read the stemplot's first row
Why: the digits survive
Figure (svg): Example 2.1's stemplot with the single leaf on row 3 boxed
\[ 3 \times 10 = 30 \]
Multiply the stem by ten
Why: a row covers ten marks
\[ 30 + \textcolor{#1f5fbf}{3} = \textcolor{#1f5fbf}{33} \]
Add the leaf
Why: the leaf supplies the ones digit
\[ \text{the list's first cell} = \textcolor{#1f5fbf}{33} \]
Check: the list's first cell
Why: the lowest leaf agrees
Faded example
Figure (svg): Try It 2.1's 30 basketball scores in cells, smallest first, from 32 to 61
Try It 2.1: 30 Park City game scores, sorted.
Fill in the blanks
the 40s row holds 12 scores; the four rows hold 30
Why: The scores in the 40s are 40, 42, 42, 43, 44, 46, 47, 47, 48, 48, 48 and 49: twelve leaves on row 4. The four rows hold 5, 12, 11 and 2 scores, and 5 + 12 + 11 + 2 = 30, one per game.
Worked example
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 3 \mid \textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{8} \to 5 \]
File the five 30s scores
Why: the lowest ten opens it
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 4 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{9} \to 12 \]
File the twelve 40s scores
Why: each ten owns one row
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 5 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{1}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{7} \to 11 \]
File the eleven 50s scores
Why: the busiest row yet
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 6 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{1} \to 2 \]
File the two 60s scores
Why: no score reaches the 70s
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 5 + 12 + 11 + 2 = 30 \]
Check: add the four rows
Why: the team played 30 games
Section
Idea 2 of 4
Concept
Figure (svg): Example 2.2's 21 supermarket distances in cells, smallest first, from 1.1 to 12.3 km
Example 2.2: distances from one home, in km.
Discussion prompt
Is any of these 21 supermarkets far enough away to stand apart from the rest?
Answer:
The list buries it; a picture that keeps the gaps will not.
Worked example
Figure (svg): Two bars on a count axis from 0 to 20: distances under 5 km, 16 of them, and 5 km or more, 5 of them
\[ \textcolor{#1f5fbf}{1.1} \ldots \textcolor{#1f5fbf}{4.8} \to 16 \]
Count distances under 5 km
Why: a bar needs counts, not values
Figure (svg): Two bars on a count axis from 0 to 20: distances under 5 km, 16 of them, and 5 km or more, 5 of them
\[ \textcolor{#1f5fbf}{5.5},\ \textcolor{#1f5fbf}{5.6},\ \textcolor{#1f5fbf}{6.5},\ \textcolor{#1f5fbf}{6.7},\ \textcolor{#1f5fbf}{12.3} \to 5 \]
Count distances from 5 km up
Why: five shops share a single bar
Figure (svg): Two bars on a count axis from 0 to 20: distances under 5 km, 16 of them, and 5 km or more, 5 of them
\[ 16 + 5 = 21 \]
Add the two counts
Why: no supermarket escapes the bins
\[ \textcolor{#1f5fbf}{12.3} - \textcolor{#1f5fbf}{6.7} = \textcolor{#1f5fbf}{5.6} \]
Subtract the two largest
Why: sizes the jump to the far shop
\[ \textcolor{#1f5fbf}{5.6} > 5 \]
Check the jump against a bin
Why: the leap outruns a whole bar
Worked example
Figure (svg): Example 2.2's 21 supermarket distances in cells, smallest first, from 1.1 to 12.3 km
\[ \textcolor{#1f5fbf}{1.1}: \text{final digit} = \textcolor{#1f5fbf}{1} \]
Take 1.1's final digit
Why: the leaf rule covers any value
Figure (svg): Distances cut into a boxed whole-kilometre stem and a boxed tenths leaf
\[ \text{the rest} = 1 \]
Keep the 1 as the stem
Why: whole kilometres group the shops
\[ \textcolor{#1f5fbf}{3.3} \to 3 \mid \textcolor{#1f5fbf}{3} \]
Cut 3.3 the same way
Why: the point moves, the rule holds
Figure (svg): Distances cut into a boxed whole-kilometre stem and a boxed tenths leaf
\[ 1 \mid \textcolor{#1f5fbf}{1}\ \textcolor{#1f5fbf}{5}, \quad 2 \mid \textcolor{#1f5fbf}{3}\ \textcolor{#1f5fbf}{5}\ \textcolor{#1f5fbf}{7} \]
File the first five shops
Why: rows run by whole kilometres
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ 1 \mid \textcolor{#1f5fbf}{5} \to \textcolor{#1f5fbf}{1.5} \]
Check: read row 1's second leaf
Why: the tenths digit came back
Worked example
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ 3 \mid \textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{8}; \ \ 4 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{8} \]
File kilometres 3 and 4
Why: kilometres pick the rows
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ 5 \mid \textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{6}; \ \ 6 \mid \textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7} \]
File the last four near shops
Why: no shop under 7 km is left
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ 7,\ 8,\ 9,\ 10,\ 11 \mid \quad \]
Write the five empty stems
Why: a stem exists without data
Figure (svg): Example 2.2's stemplot with the five blank stems 7 to 11 shaded
\[ 12 \mid \textcolor{#1f5fbf}{3} \to 1 \]
File 12.3 above them
Why: its distance from the rest shows
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ 2 + 3 + 5 + 6 + 2 + 2 + 1 = 21 \]
Check: add every row length
Why: all 21 supermarkets are here
Worked example
Figure (svg): Example 2.2's stemplot: whole kilometres 1 to 12 as stems, with 7 to 11 left blank
\[ \textcolor{#1f5fbf}{5} + \textcolor{#1f5fbf}{6} = \textcolor{#1f5fbf}{11} \]
Add rows 3 and 4
Why: the two fullest rows together
Figure (svg): Example 2.2's stemplot with rows 3 and 4 shaded
\[ 21 - \textcolor{#1f5fbf}{11} = \textcolor{#1f5fbf}{10} \]
Subtract that from 21
Why: how many shops lie elsewhere
\[ \textcolor{#1f5fbf}{11} > \textcolor{#1f5fbf}{10} \]
Compare the two groups
Why: most shops sit in two kilometres
\[ 7,\ 8,\ 9,\ 10,\ 11 \to 5 \text{ blank rows} \]
Count the blank rows
Why: the plot measures 12.3's isolation
Figure (svg): Example 2.2's stemplot with the five blank rows shaded
A value set apart: an extreme value (outlier).
\[ 5 \times 1 = 5 \text{ km} \]
Check: one km per blank row
Why: five km of road held no shop
Trap
Figure (svg): The 21 shops plotted against their row number in the shortened plot, 0 to 8: the last shop lands at 7.3, next to the cluster
\[ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 12 \]
List only the stems with leaves
Why: blank rows look like wasted paper
\[ 6 \mid \textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7}; \ \ 12 \mid \textcolor{#1f5fbf}{3} \]
Write row 12 after row 6
Why: fails: the leap is squeezed out
Figure (svg): The same 21 shops plotted against distance in km, 0 to 13, with the stretch from 7 to 12 shaded and marked no shop
\[ 7,\ 8,\ 9,\ 10,\ 11 \text{ kept blank} \]
Keep every stem in between
Why: the rows must work as a ruler
\[ \text{rows } 7 \text{ to } 11 \to 5 \]
Check: count the skipped rows
Why: five kilometres of ruler were deleted
Prediction
Figure (svg): Try It 2.2's 30 distances in cells, smallest first, from 0.5 to 8.0 miles
Predict first
Try It 2.2: 30 distances in miles, the largest of them 8.0.
Which reading would show that 8.0 stands apart?
Correct: Two blank stems sit between 5.8 and 8.0
Why: Every data set has a largest value, so being largest proves nothing. Blank rows 6 and 7 say that no student lives 6 to 8 miles out, so 8.0 breaks off from the rest instead of continuing them.
Worked example
Figure (svg): Try It 2.2's stemplot: whole miles 0 to 8 as stems, with 6 and 7 left blank
\[ 0 \mid \textcolor{#1f5fbf}{5}\ \textcolor{#1f5fbf}{7} \to 2 \]
File the shortest two
Why: row 0 is under a mile
Figure (svg): Try It 2.2's stemplot: whole miles 0 to 8 as stems, with 6 and 7 left blank
\[ 1 \mid \textcolor{#1f5fbf}{1}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{8}\,\textcolor{#1f5fbf}{9} \to 11 \]
File row 1's walks
Why: one row per whole mile
Figure (svg): Try It 2.2's stemplot: whole miles 0 to 8 as stems, with 6 and 7 left blank
\[ 2,\ 3,\ 4,\ 5 \to 7,\ 2,\ 3,\ 4 \]
Count the next four rows
Why: shape lives in the lengths
Figure (svg): Try It 2.2's stemplot: whole miles 0 to 8 as stems, with 6 and 7 left blank
\[ 6,\ 7 \mid \quad \text{then } 8 \mid \textcolor{#1f5fbf}{0} \]
File 8.0 above the blanks
Why: they are the evidence
Figure (svg): Try It 2.2's stemplot with the blank rows 6 and 7 shaded
\[ 2 + 11 + 7 + 2 + 3 + 4 + 1 = 30 \]
Check: add every row
Why: all 30 students are here
Concept
Figure (svg): An empty stem column, stems 1 to 6, ready for two sets of leaves
\[ \begin{array}{ll} \text{wins} & 25,\ 33,\ 35,\ 28,\ 13, \\ & 26,\ 30,\ 37,\ 47,\ 53 \\ \text{losses} & 57,\ 49,\ 47,\ 54,\ 69, \\ & 56,\ 52,\ 45,\ 35,\ 29 \end{array} \]
Discussion prompt
How could one stem column carry both sets without mixing them up?
Answer:
Send one set's leaves left of the stems and the other's right.
Worked example
Figure (svg): An empty stem column, stems 1 to 6, ready for two sets of leaves
\[ \textcolor{#1f5fbf}{53} \to 5 \mid \textcolor{#1f5fbf}{3} \]
Cut the last season's wins
Why: the leaf rule ignores layout
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ \textcolor{#1f5fbf}{29} \to 2 \mid \textcolor{#1f5fbf}{9} \]
Cut the same season's losses
Why: one season marks both sides
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ 5 \mid \textcolor{#1f5fbf}{3} \ \text{left}, \ \ 2 \mid \textcolor{#1f5fbf}{9} \ \text{right} \]
Send wins left and losses right
Why: the sides keep two records apart
\[ \textcolor{#1f5fbf}{13},\ \textcolor{#1f5fbf}{69} \to 1 \mid \textcolor{#1f5fbf}{3}, \ \ 6 \mid \textcolor{#1f5fbf}{9} \]
File the worst season's pair
Why: both extremes, one year
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ \textcolor{#1f5fbf}{13} - \textcolor{#1f5fbf}{3} = 10 \]
Check: take the leaf off 13
Why: a stem is worth ten games
Worked example
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\( {5 \mid 3},\ \allowbreak \allowbreak {2 \mid 9},\ \allowbreak \allowbreak {1 \mid 3},\ \allowbreak \allowbreak {6 \mid 9} \)
\[ 2 \mid \textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{8} \]
File the three 20s wins
Why: the stem digit picks the row
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ 3 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7}; \ 4 \mid \textcolor{#1f5fbf}{7} \]
File the remaining wins
Why: every season owes a win leaf
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ 3 \mid \textcolor{#1f5fbf}{5}; \ 4 \mid \textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7}\,\textcolor{#1f5fbf}{9}; \ 5 \mid \textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{6}\,\textcolor{#1f5fbf}{7} \]
File every remaining loss
Why: every season owes a loss leaf
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ \begin{aligned} \text{wins} \ \ & 1 + 3 + 4 + 1 + 1 = 10 \\ \text{losses} \ & 1 + 1 + 3 + 4 + 1 = 10 \end{aligned} \]
Check: count each side
Why: ten seasons each way
Worked example
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ \textcolor{#1f5fbf}{13} + \textcolor{#1f5fbf}{69} = 82 \]
Add the worst season's pair
Why: every game ends win or loss
\[ \textcolor{#1f5fbf}{53} + \textcolor{#1f5fbf}{29} = 82 \]
Add the best season's pair
Why: tests whether 82 was a coincidence
\[ \textcolor{#1f5fbf}{35} + \textcolor{#1f5fbf}{47} = 82 \]
Add a third season's pair
Why: a fixed schedule would explain it
\[ 82 - \textcolor{#1f5fbf}{25} = \textcolor{#1f5fbf}{57} \]
Take season 33's wins off 82
Why: predicts its losses in advance
Figure (svg): The side-by-side plot with row 5 shaded
\[ 5 \mid \textcolor{#1f5fbf}{7} \text{ on the right} = \textcolor{#1f5fbf}{57} \]
Check: hunt 57 on the right
Why: the plot holds the prediction
Faded example
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
Try It 2.3: the ten seasons, side by side.
Fill in the blanks
the fullest wins row is stem 3; it carries 4 leaves
Why: The wins side reads 1 | 3, then 5 6 8 on row 2, then 0 3 5 7 on row 3, then a single 7 on row 4 and a single 3 on row 5. Row 3 is the only one with four leaves, so the team most often won 30-something games.
Worked example
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ 1,\ 3,\ 4,\ 1,\ 1 \]
Count the wins leaves per row
Why: counts make rows comparable
Figure (svg): The side-by-side plot with each row's leaf counts printed at both edges
\[ 4 > 3 > 1 \]
Rank those row lengths
Why: the fullest row marks the commonest ten
\[ 3 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{5}\,\textcolor{#1f5fbf}{7} \]
Read row 3's four leaves
Why: the plot kept each season's total
Figure (svg): The side-by-side plot with counts printed and row 3 shaded
\[ \to \textcolor{#1f5fbf}{30},\ \textcolor{#1f5fbf}{33},\ \textcolor{#1f5fbf}{35},\ \textcolor{#1f5fbf}{37} \]
Rebuild the four totals
Why: stem and leaf give whole numbers
\[ 1 + 3 + 4 + 1 + 1 = 10 \]
Check: add the wins rows
Why: ten seasons, counted once
Section
Idea 3 of 4
Concept
Figure (svg): An empty numbered grid for a line graph: times reminded 0 to 5 across, parents 0 to 15 up
\[ \begin{array}{c|cccccc} \text{times reminded} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text{parents} & 2 & 5 & 8 & 14 & 7 & 4 \end{array} \]
Discussion prompt
Does the number of parents climb all the way, or turn somewhere along the row?
Answer:
The table hides the turn; six plotted points will not.
Worked example
Figure (svg): An empty stemplot with one stem, 0, ready for leaves
\[ \textcolor{#1f5fbf}{0},\ \textcolor{#1f5fbf}{1},\ \textcolor{#1f5fbf}{2},\ \textcolor{#1f5fbf}{3},\ \textcolor{#1f5fbf}{4},\ \textcolor{#1f5fbf}{5} \]
List the answers given
Why: every answer is one digit
\[ \text{stem} = 0, \quad \text{leaf} = \text{the digit} \]
Apply the leaf rule to 3
Why: no tens to file under
\[ 0 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{1} \ldots \to 40 \]
File all forty answers
Why: one row for the survey
Figure (svg): A stemplot with a single row, stem 0, carrying forty leaves and running off the edge
\[ \text{one row} \to \text{no ups or downs} \]
Look for a turn there
Why: fails: no shape at all
\[ 2 + 5 + 8 + 14 + 7 + 4 = 40 \]
Check: add the six counts
Why: the row holds 40
Worked example
Figure (svg): An empty numbered grid: times reminded 0 to 5 across, parents 0 to 15 up
\[ (\textcolor{#1f5fbf}{0},\ 2) \]
Plot the first pair
Why: across by answer, up by count
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ (\textcolor{#1f5fbf}{1},\ 5),\ (\textcolor{#1f5fbf}{2},\ 8) \]
Plot the next two pairs
Why: one step per extra reminder
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ (\textcolor{#1f5fbf}{3},\ 14),\ (\textcolor{#1f5fbf}{4},\ 7),\ (\textcolor{#1f5fbf}{5},\ 4) \]
Plot the last three pairs
Why: no answer may be left off
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ (\textcolor{#1f5fbf}{0},2) - (\textcolor{#1f5fbf}{1},5) - \ldots \]
Join neighbouring points
Why: a segment shows one change
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ \text{tallest point} = 14 \text{ at } \textcolor{#1f5fbf}{3} \]
Check: read the tallest point
Why: 14 is the table's biggest
Worked example
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ 14 - 8 = 6 \]
Subtract 8 from 14
Why: measures one segment's climb
Figure (svg): The chores line graph with the segment from 2 to 3 labelled plus 6
\[ \textcolor{#1f5fbf}{3} - \textcolor{#1f5fbf}{2} = 1 \]
Subtract 2 from 3
Why: measures how far across it ran
\[ 6 \div 1 = 6 \]
Divide the climb by the run
Why: parents gained per extra reminder
\[ \text{steepness} = \textcolor{#1f5fbf}{\text{change}} \div \textcolor{#1f5fbf}{\text{steps}} \]
State the rule for any segment
Why: one number compares runs of unequal length
\[ 6 \times 1 = 6 \]
Check: undo the division
Why: one step returns the whole climb
Worked example
Figure (svg): A line graph of Example 2.4: parents against times a teenager is reminded, 0 to 5 across and 0 to 15 up
\[ 7 - 14 = -7 \]
Subtract 14 from 7
Why: measures the line's first drop
Figure (svg): The chores line graph with the 2-to-3 segment labelled plus 6 and the 3-to-4 segment labelled minus 7
\[ -7 \div 1 = -7 \]
Divide the drop by one step
Why: puts it on the steepness scale
\[ 7 > 6 \]
Compare the drop with the climb
Why: interest falls quicker than it built
\[ \textcolor{#1f5fbf}{3}: 14 \text{ of the } 40 \]
Read the turn's height
Why: the commonest answer in the survey
Figure (svg): The chores line graph with the peak at 3 labelled 14 parents
\[ 14 > 8, \quad 14 > 7 \]
Check the turn against both neighbours
Why: neither side of 3 reaches 14
Worked example
Figure (svg): An empty numbered grid: seasons 33 to 42 across, wins 0 to 60 up
\[ (33, \textcolor{#1f5fbf}{25}) \ldots (37, \textcolor{#1f5fbf}{13}) \]
Plot seasons 33 to 37
Why: one point per season, in order
Figure (svg): A line graph of the Hawks' wins against season number, seasons 33 to 42, 0 to 60 wins up the side
\[ (38, \textcolor{#1f5fbf}{26}) \ldots (42, \textcolor{#1f5fbf}{53}) \]
Plot seasons 38 to 42
Why: the axis must hold every season
Figure (svg): A line graph of the Hawks' wins against season number, seasons 33 to 42, 0 to 60 wins up the side
\[ \textcolor{#1f5fbf}{13} - \textcolor{#1f5fbf}{28} = -15 \]
Subtract season 36 from 37
Why: sizes the collapse
Figure (svg): The Hawks wins line graph with the season 36 to 37 segment labelled minus 15
\[ \textcolor{#1f5fbf}{26} - \textcolor{#1f5fbf}{13} = 13 \]
Subtract season 37 from 38
Why: sizes the bounce back
Figure (svg): The wins line graph with the 36-to-37 segment labelled minus 15 and the 37-to-38 segment labelled plus 13
\[ \textcolor{#1f5fbf}{13} + 13 = \textcolor{#1f5fbf}{26} \]
Check: add the bounce onto 13
Why: season 38's total returns
Trap
Figure (svg): A line graph of four sampled seasons with the ticks equally spaced: 48 wins, then 41, 41 and 53
\[ \textcolor{#1f5fbf}{48} - \textcolor{#1f5fbf}{41} = 7, \quad \textcolor{#1f5fbf}{53} - \textcolor{#1f5fbf}{41} = 12 \]
Read both segments
Why: the two changes the eye compares
\[ 12 > 7 \]
Call the rise faster
Why: fails: spans differ
Figure (svg): The same four seasons with the axis spaced by season number, so the last segment stretches over twenty seasons
\[ 10 - 1 = 9, \quad 42 - 22 = 20 \]
Count each span
Why: the axis measures time
\[ 7 \div 9 \approx 0.78 > 0.6 = 12 \div 20 \]
Check per season
Why: the fall was faster
Prediction
Figure (svg): An empty numbered grid: repairs 0 to 3 across, people 0 to 15 up
Predict first
Try It 2.4: 40 people said how many times a year their car went in — 0 for seven of them, 1 for ten, 2 for fourteen, 3 for nine.
What must be true of the four plotted heights?
Correct: They add to 40
Why: Each person gave exactly one answer, so the four counts are the whole group shared out and nothing else. The heights need not climb or peak at the end; here the count turns downward after 2 repairs.
Worked example
Figure (svg): An empty numbered grid: repairs 0 to 3 across, people 0 to 15 up
\[ (\textcolor{#1f5fbf}{0},\ 7),\ (\textcolor{#1f5fbf}{1},\ 10) \]
Plot the first two answers
Why: across by repairs, up by people
Figure (svg): A line graph of Try It 2.4: people against times a car went in for repairs, 0 to 3 across and 0 to 15 up
\[ (\textcolor{#1f5fbf}{2},\ 14),\ (\textcolor{#1f5fbf}{3},\ 9) \]
Plot the last two answers
Why: the count turns downward after 2
Figure (svg): A line graph of Try It 2.4: people against times a car went in for repairs, 0 to 3 across and 0 to 15 up
\[ 7 + 10 = 17 \]
Add the first two heights
Why: a running total of people asked
\[ 17 + 14 + 9 = 40 \]
Add the last two heights
Why: everybody answered exactly once
\[ 40 - 7 - 10 - 14 - 9 = 0 \]
Check: take each height off 40
Why: nothing left over, nothing short
Faded example
Figure (svg): A line graph of Try It 2.4: people against times a car went in for repairs, 0 to 3 across and 0 to 15 up
Try It 2.4: the finished line graph.
Fill in the blanks
changes across the three steps: 3, 4, −5; steepest, in size = 5 per step
Why: Reading the heights in order, 10 − 7 = 3, then 14 − 10 = 4, then 9 − 14 = −5. Every step is one repair wide, so each change is already its steepness, and the −5 drop is the biggest in size, so it is the steepest.
Worked example
Figure (svg): A line graph of Try It 2.4: people against times a car went in for repairs, 0 to 3 across and 0 to 15 up
\[ \textcolor{#1f5fbf}{10} - \textcolor{#1f5fbf}{7} = 3 \]
Subtract the first two heights
Why: the line's opening climb
Figure (svg): The repairs line graph with the first segment labelled plus 3
\[ \textcolor{#1f5fbf}{14} - \textcolor{#1f5fbf}{10} = 4 \]
Subtract the next two heights
Why: a second climb to compare
Figure (svg): The repairs line graph with the first two segments labelled plus 3 and plus 4
\[ \textcolor{#1f5fbf}{9} - \textcolor{#1f5fbf}{14} = -5 \]
Subtract the last two heights
Why: the only fall on this line
Figure (svg): The repairs line graph with all three segments labelled plus 3, plus 4 and minus 5
\[ 5 \div 1 = 5 \]
Divide the fall by one step
Why: people lost per extra repair
\[ \textcolor{#1f5fbf}{7} + 3 + 4 - 5 = \textcolor{#1f5fbf}{9} \]
Check: walk the changes from 7
Why: arriving at the last height, 9
Section
Idea 4 of 4
Concept
Figure (svg): Four bars drawn to scale for the 2030 graduates: Northeast 517,720, Midwest 695,170, South 1,253,540, West 749,400
Example 2.6: graduates projected for 2030.
Discussion prompt
Does the South on its own expect more than half the country's graduates?
Answer:
Doubling the South's count settles it; the whole has to be built first.
Worked example
Figure (svg): Four bars drawn to scale for the 2030 graduates: Northeast 517,720, Midwest 695,170, South 1,253,540, West 749,400
\[ \textcolor{#1f5fbf}{1{,}253{,}540} - \textcolor{#1f5fbf}{517{,}720} = \textcolor{#1f5fbf}{735{,}820} \]
Subtract the Northeast from the South
Why: sizes the distance between two bars
\[ \textcolor{#1f5fbf}{735{,}820} > \textcolor{#1f5fbf}{517{,}720} \]
Compare that gap with the Northeast
Why: the gap alone outruns a whole region
\[ \textcolor{#1f5fbf}{735{,}820} \text{ out of what?} \]
Ask what the gap is part of
Why: fails: no whole has been counted
\[ \textcolor{#1f5fbf}{517{,}720} + \textcolor{#1f5fbf}{735{,}820} = \textcolor{#1f5fbf}{1{,}253{,}540} \]
Check: add the gap back on
Why: the South's own count returns
Worked example
Figure (svg): The four regions' graduate counts stacked into one column against an axis marked in millions
\[ \textcolor{#1f5fbf}{517{,}720} + \textcolor{#1f5fbf}{695{,}170} = \textcolor{#1f5fbf}{1{,}212{,}890} \]
Add the first two counts
Why: stacking regions builds the whole
Figure (svg): The four regions' graduate counts stacked into one column against an axis marked in millions
\[ \textcolor{#1f5fbf}{1{,}212{,}890} + \textcolor{#1f5fbf}{1{,}253{,}540} = \textcolor{#1f5fbf}{2{,}466{,}430} \]
Add the South's count
Why: no region may be left out
Figure (svg): The four regions' graduate counts stacked into one column against an axis marked in millions
\[ \textcolor{#1f5fbf}{2{,}466{,}430} + \textcolor{#1f5fbf}{749{,}400} = \textcolor{#1f5fbf}{3{,}215{,}830} \]
Add the West's count
Why: every graduate counted once
Figure (svg): The four regions' graduate counts stacked into one column against an axis marked in millions
\[ \textcolor{#1f5fbf}{3{,}215{,}830} - \textcolor{#1f5fbf}{749{,}400} = \textcolor{#1f5fbf}{2{,}466{,}430} \]
Check: take the West back off
Why: the three-region total returns
Worked example
Figure (svg): The four regions' graduate counts stacked into one column against an axis marked in millions, with the full column marked 3,215,830 in all
\[ \textcolor{#1f5fbf}{517{,}720} \div \textcolor{#6b7280}{3{,}215{,}830} \approx 0.161 \]
Divide Northeast by the whole
Why: what part it fills
\[ 0.161 \times 100 = \textcolor{#b54708}{16.1} \]
Multiply that fraction by 100
Why: now per hundred
\[ \textcolor{#b54708}{16.1\%} \text{ of the column} \]
Mark it on a per-cent column
Why: height means share
Figure (svg): The four regions as shares of one column against a per-cent axis from 0 to 100
\[ \textcolor{#b54708}{\text{share}} = \frac{\textcolor{#1f5fbf}{\text{part}}}{\textcolor{#6b7280}{\text{whole}}} \times 100 \]
State the rule for any part
Why: works on any data
\[ \textcolor{#b54708}{16.1} < 25 \]
Check the segment against 25
Why: it fills under a quarter
Worked example
Figure (svg): The four regions as shares of one column against a per-cent axis from 0 to 100
\[ \textcolor{#1f5fbf}{695{,}170} \div \textcolor{#6b7280}{3{,}215{,}830} \approx 0.216 \]
Divide Midwest by the whole
Why: the base must not change
\[ \begin{aligned} \textcolor{#1f5fbf}{1{,}253{,}540} &\div \textcolor{#6b7280}{3{,}215{,}830} \approx 0.390 \\ \textcolor{#1f5fbf}{749{,}400} &\div \textcolor{#6b7280}{3{,}215{,}830} \approx 0.233 \end{aligned} \]
Divide the last two counts
Why: every region needs a share
\[ 0.216,\ 0.390,\ 0.233 \to \textcolor{#b54708}{21.6},\ \textcolor{#b54708}{39.0},\ \textcolor{#b54708}{23.3} \]
Rescale all three
Why: shares now compare
Figure (svg): The four regions as shares of one column against a per-cent axis from 0 to 100
\[ \textcolor{#b54708}{16.1} + \textcolor{#b54708}{21.6} + \textcolor{#b54708}{39.0} + \textcolor{#b54708}{23.3} = \textcolor{#b54708}{100.0} \]
Check: add the shares
Why: one whole, no more
Worked example
Figure (svg): Four bars for the regions' shares on a per-cent axis from 0 to 50, with a grey line at 50: Northeast 16.1, Midwest 21.6, South 39.0, West 23.3
\[ \textcolor{#b54708}{39.0} > \textcolor{#b54708}{23.3} > \textcolor{#b54708}{21.6} > \textcolor{#b54708}{16.1} \]
Rank the four shares
Why: bar lengths order the regions
\[ \textcolor{#b54708}{39.0} < 50 \]
Compare the tallest bar with 50
Why: no region holds a majority
\[ \textcolor{#b54708}{39.0} + \textcolor{#b54708}{23.3} = \textcolor{#b54708}{62.3} \]
Add the two tallest shares
Why: tests whether two regions can
\[ \textcolor{#b54708}{62.3} > 50 \]
Compare their total with 50
Why: South and West pass half
\[ \textcolor{#b54708}{100.0} - \textcolor{#b54708}{62.3} = \textcolor{#b54708}{37.7} \]
Check: take the pair off 100
Why: 16.1 and 21.6 add to 37.7
Trap
Figure (svg): Districts 1 and 3 on one per-cent axis, each twice: district 1 is 15.5% of the registered voters and 19.4% of all residents; district 3 is 9.8% of the voters and 9.0% of the residents
\[ \textcolor{#b54708}{19.4} - \textcolor{#b54708}{15.5} = \textcolor{#b54708}{3.9} \]
Subtract 15.5 from 19.4
Why: both look like per cents
\[ \textcolor{#b54708}{9.0} - \textcolor{#b54708}{9.8} = \textcolor{#b54708}{-0.8} \]
Try district 3 too
Why: fails: now negative
Figure (svg): The same four districts on the same scale, now as shares of all residents: 19.4, 15.6, 9.0 and 18.5
\[ \textcolor{#b54708}{19.4} + \textcolor{#b54708}{15.6} + \textcolor{#b54708}{9.0} + \textcolor{#b54708}{18.5} = \textcolor{#b54708}{62.5} \]
Add the residents column
Why: its own whole
\[ \textcolor{#b54708}{62.5} \ne 100 \]
Check against 100
Why: two are missing
Prediction
Figure (svg): Four bars for districts 1 to 4's share of registered voters on a per-cent axis from 0 to 20: 15.5, 12.2, 9.8 and 17.4
Predict first
Try It 2.6: districts 1 to 4 hold 15.5%, 12.2%, 9.8% and 17.4% of the city's registered voters. Districts 5 and 6 are not shown.
What can be said about the two missing districts?
Correct: Together they hold 45.1% of registered voters
Why: Shares of one whole add to 100%, so the districts not listed hold whatever the four listed ones do not. How that remainder splits between the two is still unknown, and no head counts are needed to get the pair's total.
Worked example
Figure (svg): Four bars for districts 1 to 4's share of registered voters on a per-cent axis from 0 to 20: 15.5, 12.2, 9.8 and 17.4
\[ \textcolor{#b54708}{15.5} + \textcolor{#b54708}{12.2} + \textcolor{#b54708}{9.8} + \textcolor{#b54708}{17.4} = \textcolor{#b54708}{54.9} \]
Add the four listed shares
Why: how much of the whole is shown
Figure (svg): The registered-voter bars with a grey line at 54.9%, the four listed districts' total
\[ \textcolor{#b54708}{100} - \textcolor{#b54708}{54.9} = \textcolor{#b54708}{45.1} \]
Take that total off 100
Why: what the unlisted pair holds
\[ \textcolor{#b54708}{45.1} > \textcolor{#b54708}{17.4} \]
Compare with the largest listed
Why: the pair beats any district
\[ \textcolor{#b54708}{17.4} + \textcolor{#b54708}{15.5} + \textcolor{#b54708}{12.2} + \textcolor{#b54708}{9.8} = \textcolor{#b54708}{54.9} \]
Check: add the shares largest first
Why: a total ignores the order
Faded example
Figure (svg): Three bars drawn to scale for Park City's age groups: children 67,059, working-age adults 152,198, retirees 131,662
Try It 2.5: the town, counted by age group.
Fill in the blanks
the town holds 350919 people; children = 19.1%; retirees = 37.5%
Why: 67,059 + 152,198 + 131,662 = 350,919. Then 67,059 ÷ 350,919 ≈ 0.191, which is 19.1%, and 131,662 ÷ 350,919 ≈ 0.375, which is 37.5%. The book rounds these to 19% and 38%.
Worked example
Figure (svg): Three bars drawn to scale for Park City's age groups: children 67,059, working-age adults 152,198, retirees 131,662
\[ \begin{aligned} &\textcolor{#1f5fbf}{67{,}059} + \textcolor{#1f5fbf}{152{,}198} \\ &{+}\ \textcolor{#1f5fbf}{131{,}662} = \textcolor{#6b7280}{350{,}919} \end{aligned} \]
Add the three counts
Why: one town total
\[ \begin{aligned} \textcolor{#1f5fbf}{67{,}059} &\div \textcolor{#6b7280}{350{,}919} \approx 0.191 \\ \textcolor{#1f5fbf}{152{,}198} &\div \textcolor{#6b7280}{350{,}919} \approx 0.434 \\ \textcolor{#1f5fbf}{131{,}662} &\div \textcolor{#6b7280}{350{,}919} \approx 0.375 \end{aligned} \]
Divide each by the town
Why: one shared whole
Figure (svg): Park City's three age groups as shares of one column against a per-cent axis from 0 to 100
\[ 0.191,\ 0.434,\ 0.375 \to \textcolor{#b54708}{19.1},\ \textcolor{#b54708}{43.4},\ \textcolor{#b54708}{37.5} \]
Rescale to per cents
Why: the column's own axis
\[ \textcolor{#b54708}{19.1} + \textcolor{#b54708}{43.4} + \textcolor{#b54708}{37.5} = \textcolor{#b54708}{100.0} \]
Check: add the shares
Why: the column fills
Pattern
\[ \text{value} = \text{stem} \times 10 + \text{leaf}, \qquad \textcolor{#b54708}{\text{share}} = \frac{\textcolor{#1f5fbf}{\text{part}}}{\textcolor{#6b7280}{\text{whole}}} \times 100 \]
Check
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
Check your understanding
From Try It 2.1's stemplot, in how many of the 30 games did Park City score fewer than 45 points?
Answer: A
Why: Row 3 holds 32, 32, 33, 34 and 38, all of them under 45, which is five games. Row 4 holds twelve scores, but only 40, 42, 42, 43 and 44 are under 45: five more. 5 + 5 = 10.
Worked example
Figure (svg): Try It 2.1's stemplot: stems 3 to 6 holding the 30 basketball scores
\[ 3 \mid \textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4}\,\textcolor{#1f5fbf}{8} \to 5 \]
Count all of row 3
Why: every 30s score is under 45
Figure (svg): Try It 2.1's stemplot with row 3 shaded
\[ 4 \mid \textcolor{#1f5fbf}{0}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{2}\,\textcolor{#1f5fbf}{3}\,\textcolor{#1f5fbf}{4} \to 5 \]
Count row 4's leaves below 5
Why: 40 to 44 stay under 45
Figure (svg): Try It 2.1's stemplot with the first five leaves of row 4 boxed
\[ 5 + 5 = 10 \]
Add the two counts
Why: all games under the cut-off
\[ 12 - 5 = 7 \]
Subtract from row 4's length
Why: splits row 4 at the cut-off
\[ 10 + 7 + 11 + 2 = 30 \]
Check: add the four groups
Why: the season's 30 games, all counted
Check
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
Check your understanding
A fan says the side-by-side plot proves the Hawks improved, because the losses side sits higher up the stems. Is that reasoning sound?
Answer: A
Why: A side-by-side stemplot sorts each set into rows and throws away which season each value came from. It can say losses ran higher than wins across these ten years; it cannot say whether the team got better or worse as the seasons passed.
Worked example
Figure (svg): A side-by-side stemplot: the Hawks' wins as leaves left of the stem, losses right, seasons 33 to 42
\[ \text{wins side: } \textcolor{#1f5fbf}{13} \text{ to } \textcolor{#1f5fbf}{53} \]
Read the wins side's ends
Why: the plot stores values, not dates
\[ \textcolor{#1f5fbf}{25} \text{ in season } 33, \quad \textcolor{#1f5fbf}{53} \text{ in season } 42 \]
Name the seasons behind them
Why: the rows never recorded a year
Figure (svg): A line graph of the Hawks' wins against season number, seasons 33 to 42, 0 to 60 wins up the side
\[ \textcolor{#1f5fbf}{53} - \textcolor{#1f5fbf}{25} = \textcolor{#1f5fbf}{28} \]
Subtract season 33 from 42
Why: the climb no row could show
\[ 42 - 33 = 9 \]
Count the steps between
Why: nine seasons carried the gain
\[ \textcolor{#1f5fbf}{28} \div 9 \approx 3.1 \]
Check: share the gain over nine
Why: about three extra wins a season
Worked example
Figure (svg): Example 2.1's stemplot: stems 3 to 10 with leaves
\[ 6 \mid \ \to \textcolor{#1f5fbf}{7}, \quad 9 \mid \ \to \textcolor{#1f5fbf}{7} \]
Find the two longest rows
Why: the tens where scores pile up
Figure (svg): Example 2.1's stemplot with rows 6 and 9 shaded
\[ \textcolor{#1f5fbf}{7} + \textcolor{#1f5fbf}{7} = \textcolor{#1f5fbf}{14} \]
Add the two row lengths
Why: how many scores sit in the peaks
\[ \textcolor{#1f5fbf}{14} \div 31 \approx 0.452 \]
Divide by the class size
Why: their share of Professor Dean's class
\[ 0.452 \to \textcolor{#b54708}{45.2\%} \]
Rescale the fraction
Why: puts the peaks per hundred
\[ \textcolor{#b54708}{45.2} < 50 \]
Check the share against a half
Why: the peaks fall just short of half
Recap
The book's Table 2.5: 40 age leaves, 39 presidents.
OpenStax Introductory Statistics 2e, §2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs §2.1, pp. 66-74 — Examples 2.1, 2.2, 2.4, 2.6 and the Try Its trace back here
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