VertexMath · MTH1W · Free worksheet

Data Management Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Frequency table mean: multiply each value by its frequency, add, then divide by the total frequency, not by the number of rows.
Median: put the data in order first. Odd count: the middle value. Even count: the mean of the two middle values.
Quartiles: $Q_1$ and $Q_3$ are the medians of the lower and upper halves; with an odd count, the median belongs to neither half. $IQR=Q_3-Q_1$ is one number. An outlier lies below $Q_1-1.5\times IQR$ or above $Q_3+1.5\times IQR$ .
Circle graph: sector angle $=$ fraction of the total $\times 360^\circ$ .
Interpolation stays inside the range of the data; extrapolation goes outside it and is less reliable. A strong correlation does not prove that one variable causes the other.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Types of Data · Mean, Median, Mode · Quartiles · Lines of Best Fit · Predictions

K1 Circle Graph

A survey asked 40 Grade 9 students how they usually get to school.

$\text{Way to school}$	Walk	Bus	Car	Bike
$\text{Students}$	$14$	$12$	$10$	$4$

a)
State whether each variable is categorical, discrete or continuous: (i) way to school, (ii) number of people in a student's household, (iii) time to get to school.
b)
Determine the percent of students in each category.
c)
Determine the sector angle of each category in a circle graph.

Stuck? Peek at a hint

Write each category as a fraction of the 40 students first.

K2 Summer Reading

The table shows how many books each of 25 students read over the summer.

$\text{Books read}$	$0$	$1$	$2$	$3$	$4$	$5$
$\text{Students}$	$2$	$9$	$5$	$4$	$3$	$2$

a)
Determine the mean number of books read.
b)
Determine the median.
c)
State the mode.

Stuck? Peek at a hint

Add the bottom row first: that is how many students, and how many data values, there are.

K3 Quartiles and Outliers

Eleven students recorded how many minutes they spent on homework last night:
$21,\ 15,\ 45,\ 18,\ 12,\ 24,\ 17,\ 26,\ 15,\ 20,\ 23$

a)
Determine the five-number summary.
b)
Determine the interquartile range.
c)
Determine the values $Q_1-1.5\times IQR$ and $Q_3+1.5\times IQR$ , then identify any outliers.

Stuck? Peek at a hint

Put the eleven values in order before you look for the middle.

K4 Screen Time and Sleep

The scatter plot shows the daily screen time and the nightly sleep, in hours, of 10 students, with a line of best fit. Each grid square is 1 hour.

a)
State the direction and the strength of the correlation.
b)
Determine the equation of the line of best fit, with $x$ the screen time and $y$ the sleep.
c)
Use your equation to determine the screen time of a student who sleeps $6.5$ hours.

Stuck? Peek at a hint

Pick two points where the line crosses grid corners, then find rise over run.

K5 Growing Sunflower

The model $h=2.5t+4$ gives the height $h$ , in centimetres, of a sunflower $t$ days after planting. It was built from measurements taken on days 10, 20, 30, 40, 50 and 60.

a)
Determine $h$ when $t=35$ . State whether this is interpolation or extrapolation.
b)
Determine $h$ when $t=90$ . State whether this is interpolation or extrapolation.
c)
Determine $t$ when the sunflower is $16.5$ cm tall. State whether this is interpolation or extrapolation.

Stuck? Peek at a hint

Compare each value of $t$ with the days that were measured. In (c), set $h=16.5$ and solve for $t$ .

Real World (A)

Combined Means · Car Prices · Comparing Box Plots

A1 Two Classes, One Test

Two Grade 9 classes write the same test. Period 1 has 30 students and a mean mark of $70\%$ . Period 2 has 20 students and a mean mark of $78\%$ .

a)
Determine the mean mark of all 50 students.
b)
The teacher finds that one Period 2 mark of $100\%$ was entered as $10\%$ . Determine the corrected Period 2 mean.
c)
A Period 1 student who scored $60\%$ rewrites the test, and the new mark replaces the old one. What mark would raise the Period 1 mean to $72\%$ ? Is that possible?

Stuck? Peek at a hint

Turn each class mean back into a class total: total $=$ mean $\times$ number of students.

A2 Used Car Prices

The table shows the age and price of eight used compact cars for sale in Ontario. A line of best fit for the data is $P=-1.8a+24$ , where $a$ is the age in years and $P$ is the price in thousands of dollars.

$\text{Age (years)}$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$\text{Price (\$1000s)}$	$22.0$	$20.5$	$18.4$	$17.0$	$15.2$	$13.0$	$11.6$	$9.5$

a)
Interpret the slope and the $P$ -intercept in this context.
b)
Predict the price of a car that is $4.5$ years old. Is this interpolation or extrapolation?
c)
What does the model predict for a 15-year-old car? What does this tell you about using the model?

Stuck? Peek at a hint

$P$ is in thousands of dollars: $P=11.4$ means $11,400.

A3 Which Bus?

Two bus routes take students from the same neighbourhood to school. The box plots show the travel time, in minutes, of each route on the same 20 school days. Each grid square is 2 minutes.

a)
Determine the median and the interquartile range of each route.
b)
A rider says most Route 7 trips take between 28 and 40 minutes, because that part of the plot is the longest. About how many of the 20 trips are in that part?
c)
A student must get to school in 30 minutes or less. On how many of the 20 days did each route do this?

Stuck? Peek at a hint

Read the five-number summary of each route off the plot first.

Think It Through (T)

Misleading Graphs · Counterexamples

T1 Spot the Flaw

A town newsletter shows the mean number of daily visitors at its two outdoor pools last summer. A student reads the graph and claims:
"Pool B gets four times as many visitors as Pool A."

a)
Determine the daily visitors at each pool, and the percent by which Pool B's number is greater than Pool A's.
b)
Explain why the graph makes Pool B look four times as large.
c)
State how to redraw the graph fairly, and the ratio of the bar heights in the fair graph.

Stuck? Peek at a hint

Read each bar's value from the vertical axis, then look at where that axis starts.

T2 Steep Is Not Strong

A student claims: "The steeper the line of best fit, the stronger the correlation."

Graph A

Graph B

a)
Write four data points that have a perfect positive correlation and a line of best fit with slope $\frac{1}{10}$ .
b)
Graphs A and B use the same scales. Which graph shows the stronger correlation? Which would have the steeper line of best fit?
c)
Explain what the strength of a correlation describes, and why the student's claim is false.

Stuck? Peek at a hint

For (a), choose points that all lie exactly on one line.

Boss Level

Optional challenge

Moving the Mean

a)
The mean of six test scores is $75$ . After a seventh test, the mean of all seven scores is $78$ . Find the seventh score.
b)
Five different positive whole numbers have a median of $10$ and a mean of $12$ . Find the largest possible value of the greatest number.
c)
For five numbers with the same conditions as in (b), find the smallest possible value of the greatest number.

Stuck? Peek at a hint

Work with totals: a mean times a count gives a sum. For (b) and (c), write the list as $a<b<10<d<e$ with $a+b+d+e=50$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) (i) categorical; (ii) discrete; (iii) continuous
b) Walk $35\%$ , Bus $30\%$ , Car $25\%$ , Bike $10\%$
c) Walk $126^\circ$ , Bus $108^\circ$ , Car $90^\circ$ , Bike $36^\circ$

a) $2.12$ books
b) $2$ books
c) $1$ book

a) Min $12$ , $Q_1=15$ , median $20$ , $Q_3=24$ , max $45$
b) $IQR=9$ minutes
c) $1.5$ and $37.5$ ; $45$ is an outlier (nothing is below $1.5$ )

a) Negative, strong
b) $y=-0.5x+10$
c) $7$ hours

a) $91.5$ cm; interpolation
b) $229$ cm; extrapolation
c) $t=5$ days; extrapolation

Real World

a) $73.2\%$
b) $82.5\%$
c) $120\%$ : impossible, since no mark is above $100\%$ . The Period 1 mean can reach at most about $71.3\%$ .

a) Slope $-1.8$ : the price drops by about $1800 for each year of age. Intercept $24$ : a new car (age 0) costs about $24,000.
b) $15,900; interpolation
c) $P=-3$ , a price of $-$ $3000: impossible. 15 years is far outside the data (1 to 8 years), so the model should not be used there.

a) Route 7: median $24$ min, $IQR=6$ min. Route 12: median $26$ min, $IQR=4$ min.
b) About $5$ trips (one quarter of the data)
c) Route 12: all $20$ days. Route 7: at least $15$ and at most $19$ days; the box plot cannot give the exact number.

Think It Through

a) Pool A: $420$ ; Pool B: $480$ ; about $14.3\%$ greater
b) The vertical axis starts at $400$ , not $0$ . The bars show only the $20$ and $80$ visitors above $400$ , and $80=4\times20$ .
c) Start the vertical axis at $0$ ; the heights are then in the ratio $480:420=8:7\approx1.14$ .

a) e.g. $(0,2)$ , $(10,3)$ , $(20,4)$ , $(30,5)$
b) Stronger correlation: A. Steeper line: B.
c) Strength describes how closely the points lie to a line, not how steep the line is. (a) and (b) are counterexamples: a shallow line can fit perfectly, and a steep one can fit poorly.

Boss Level

Boss

a) $96$
b) $36$ (the list $1, 2, 10, 11, 36$ )
c) $17$ (the list $8, 9, 10, 16, 17$ )

Free cheat sheet

Data Management Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedFind the five-number summary, the IQR and any outliers of $4,6,8,9,10,12,14,16,31$ . (card 5)

1Types of data

UsePrimary: you collect it; secondary: someone else did. A sample is part of the population.

labels or words	categorical
counted: $0,1,2,\ldots$	discrete
measured: any value	continuous

TrapTime rounded to whole minutes is still continuous: it is measured.

2Graphs

$\text{sector angle}=\dfrac{\text{part}}{\text{total}}\times360^\circ$

Which graph?

compare categories	bar graph
parts of one whole	circle graph
continuous data in intervals	histogram (bars touch)
two numerical variables	scatter plot

Example $9$ of $24$ students: $9/24\times360^\circ=135^\circ$ .

TrapA 15% sector is $0.15\times360^\circ=54^\circ$ , not $15^\circ$ .

3Mean, median, mode

$\text{mean}=\dfrac{\text{sum of the values}}{\text{number of values}}$

$\text{range}=\text{max}-\text{min}$

UseMedian: order first. Odd count: the middle value; even: the mean of the two middle values. Mode: the most frequent value.

Example $3,8,5,8,6$ ordered: $3,5,6,8,8$ . Mean $6$ , median $6$ , mode $8$ , range $5$ .

TrapValues $0,1,2$ with frequencies $3,5,2$ : mean $=(0+5+4)/10=0.9$ , not $(0+1+2)/3=1$ .

4Using the mean

$\text{total}=\text{mean}\times\text{count}$

UseAn outlier pulls the mean; the median barely moves. Combine groups by adding totals.

ExampleWages $15, $16, $16, $18, $60 an hour: mean $25, median $16.

Trap $24$ students average $65$ and $16$ average $80$ : $(24\cdot65+16\cdot80)/40=71$ , not $72.5$ .

5Quartiles and IQR

$\text{IQR}=Q_3-Q_1$

$\text{outlier}<Q_1-1.5\times\text{IQR}$

$\text{outlier}>Q_3+1.5\times\text{IQR}$

UseOrder first. $Q_1$ , $Q_3$ : medians of the lower and upper halves. Odd count: the median is in neither half.

Example $4,6,8,9,10,12,14,16,31$ : $Q_1=7$ , median $10$ , $Q_3=15$ , $\text{IQR}=8$ ; $15+12=27$ , so $31$ is an outlier.

Trap $1,3,5,7,9$ : $Q_1=2$ (median of $1,3$ ), not $3$ (median of $1,3,5$ ).

6Box plots

UseDraw from the five-number summary. Compare medians for the centre, IQRs for the spread.

TrapA long whisker, $30$ to $55$ , still holds only about a quarter of the data.

7Correlation

UseStrength: how closely the points lie to a line.

points rise left to right	positive
points fall left to right	negative
no clear pattern	no correlation

TrapSteep is not strong: points exactly on $y=0.2x+5$ are a perfect correlation.

8Line of best fit

$m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}$

$y=mx+b$

UseFollow the trend. Use two points on the line; give $m$ and $b$ with units.

ExampleThrough $(2,30)$ and $(8,12)$ : $m=-18/6=-3$ ; $30=-6+b$ , so $y=-3x+36$ .

Trap $(2,4)$ to $(8,7)$ : $m=3/6=0.5$ , not $6/3=2$ .

TrapGiven $y=21$ : solve $21=-3x+36$ , so $x=5$ ; not $-3(21)+36$ .

9Predictions

UseCorrelation is not cause: a third variable can drive both. Critique: axis not from $0$ ? Biased sample?

$x$ inside the data's range	interpolation
$x$ below or above it	extrapolation (risky)

TrapAxis from $50$ : bars $52$ and $56$ look 1 to 3; $56$ is only about 8% more.

Which average?

Symmetric data, no outliers Mean

Outliers or skewed data Median

Categories, or the most common value Mode

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