QuanticaPrealgebraOpen in the courseIn 10.6 you squeezed a whole list down to one number with the mean, median, and mode. Now we go the other way and read data that someone has already arranged for you. A good display shows in one look what a bare list hides, like the biggest category, a trend over time, or how a whole splits into parts. In this lesson you will read tables, bar and line graphs, pie charts, stem-and-leaf plots, and histograms.
Problem
The table above shows how many boxes of cookies three scout troops sold in October and November. Birch's Total is highlighted with a ? because it is missing. Complete the table by finding Birch's Total. Give the number.
Show a hint
- A Total is just the two months added together, so find Birch's row and look only at its October and November cells.
- Read across Birch's row from left to right, then add the October number to the November number to fill in the ? cell.
Show the full solution
Birch's row reads 120 in October and 65 in November, and a Total is just the two months added, so $$120 + 65 = \boxed{185}.$$
Problem
The bar graph above shows how many marbles of each color are in a jar. Which color is most common, and how many marbles of that color are there? Give the number.
Show a hint
- The most common color is the one with the tallest bar. Scan across all five bars and find the one that reaches highest.
- Once you have spotted the tallest bar, trace from its top straight over to the number scale on the left to read its height.
Show the full solution
The blue bar is the tallest of the five, so blue is the most common color. Tracing the top of that bar across to the scale gives \(\boxed{22}\) marbles.
Problem
The bar graph above shows how many goals five players scored. How many more goals did the top scorer make than the lowest scorer? Give the number.
Show a hint
- Find the tallest bar and the shortest bar. The tallest tells you the top scorer's goals, the shortest tells you the lowest scorer's goals.
- Once you have those two numbers, subtract the smaller from the larger to see how many more goals the top scorer made.
Show the full solution
The tallest bar reaches 8 and the shortest reaches 3, so $$8 - 3 = \boxed{5}.$$ Questions that ask how many more want the gap between two bars, not the height of either one.
Problem
The line graph above tracks two food trucks over five days. On exactly one day the two trucks sold the same amount, which is where the lines meet. How many did each truck sell on that day? Give the number.
Show a hint
- The two lines start apart and end apart, but there is one spot where they cross. Find the day where they touch at the same point.
- Once you spot where the lines meet, slide straight across to the vertical axis and read off the height at that shared point.
Show the full solution
The two lines cross on day 3. Sliding across from that crossing point to the vertical axis gives \(\boxed{45}\) for each truck. Where two lines cross, both quantities are equal, so one reading answers for both.
Problem
The line graph above tracks a plant's height in centimeters across six weeks. During which week did the plant climb the most from the week before, meaning the week with the steepest rise on the graph? Give the week number.
Show a hint
- The biggest growth is the steepest part of the line, where it climbs fastest from one dot to the next. Scan across the graph and find where the jump between two neighboring weeks looks largest.
- To be sure, work out each week-to-week change by subtracting the earlier height from the next one. The week you want is the one where that difference is the biggest, and you name the later week.
Show the full solution
The heights read 2, 5, 9, 16, 22, and 25, so the week-to-week gains are 3, 4, 7, 6, and 3. The biggest is the 7 centimeter jump from 9 up to 16, which is the climb into week \(\boxed{4}\). The steepest segment marks the biggest change, which is not the same as the highest point on the graph.
Problem
The pie chart above shows the favorite season of 400 surveyed students. How many of them chose summer? Give the number.
Show a hint
- Find the summer slice on the pie chart above and read the percent printed on it. That percent tells you the share of the 400 students who picked summer.
- To turn a percent of a group into a count, multiply the percent (as a decimal) by the total number of students. So take summer's percent, write it as a decimal, and multiply by 400.
Show the full solution
The summer slice is labeled \(40\%\), so take 40 percent of the 400 students, $$0.40 \times 400 = \boxed{160}.$$ A slice on its own only gives a share, so you need the group total to turn it into a count.
Problem
The pie chart above shows students' favorite pets, but the reptile slice is left blank with a question mark. What percent is the reptile slice? Give the number.
Show a hint
- Every slice of a pie chart together covers the whole circle, so all the percents have to add up to 100.
- Add the four percents the chart already labels, then see how much is left over to reach 100.
Show the full solution
The five slices have to add to 100 percent, so the missing one is whatever the four labeled slices leave behind, $$100-(35+25+15+12)=100-87=\boxed{13}.$$
Problem
The pie chart above shows the favorite season of 400 students. How many more students chose summer than winter? Give the number.
Show a hint
- Summer is the biggest slice and winter is one of the smallest. You can subtract the percents first, finding summer's percent minus winter's percent, then take that percent of 400.
- Or find each count on its own by multiplying its percent by 400, then subtract the winter count from the summer count.
Show the full solution
Summer is \(0.40 \times 400 = 160\) students and winter is \(0.15 \times 400 = 60\), so $$160 - 60 = \boxed{100}.$$ Subtracting the percents first works too, since \(25\%\) of 400 is the same 100.
Problem
The stem-and-leaf plot above shows eleven quiz scores. Find the mode of the scores. Give the number.
Show a hint
- The mode is the value that shows up most often. On a stem-and-leaf plot, that means hunting for the leaf that repeats the most on a single stem.
- Scan each stem for a leaf written more than once, then rebuild the full score by joining that stem to its repeated leaf the way the key shows.
Show the full solution
On stem 6 the leaf 8 is written three times, more often than any other leaf on any stem, so the mode is $$6\,|\,8 = \boxed{68}.$$ The most repeated leaf gives the mode directly, so there is no need to write out all eleven scores.
Problem
The histogram above shows how many households own each number of pets. How many households were surveyed in all? Give the number.
Show a hint
- Each bar's height tells you how many households fall into that pet count, so no single bar is the whole survey.
- To count everyone once, add the heights of all the bars together.
Show the full solution
Every household lands in exactly one bar, so add the bar heights, $$5+12+8+3+2=\boxed{30}.$$ The heights count households, not pets, so summing them counts everyone surveyed exactly once.
Problem
The stem-and-leaf plot above shows how many push-ups each of thirteen students did. What is the median number of push-ups? Give the number.
Show a hint
- The leaves in a stem-and-leaf plot are already listed in order from smallest to largest, so you can just count through the values as they appear from the top row down.
- With thirteen values, the median is the middle one, the 7th value counting up from the smallest. Count to the 7th leaf and read its stem and leaf together.
Show the full solution
With thirteen values the median is the 7th counting up from the smallest. The leaves are already in order, and counting from the top of the plot the 7th one sits on stem 2 with leaf 7, so $$\text{median} = \boxed{27}$$ A stem-and-leaf plot sorts the data for you, so you can count straight to the middle.
Problem
The histogram above shows how many households own each number of pets. What is the mean number of pets per household? Give your answer as a decimal.
Show a hint
- A mean is the total number of pets divided by the total number of households. Read the height of each bar to get how many homes fall in that group, and add those heights to find the total households.
- To get the total pets, multiply each pet count by the number of households at that bar, then add all those products. That handles the fact that some counts happen far more often than others.
Show the full solution
Multiply each pet count by the number of households that own that many, add those products, then divide by the \(5+12+8+3+2=30\) households, $$\frac{0\cdot5+1\cdot12+2\cdot8+3\cdot3+4\cdot2}{30}=\frac{45}{30}=\boxed{1.5}$$ A histogram groups values, so the mean has to weight each value by how often it occurs.
Problem
The bar graph above shows the daily high temperature for six days, Monday through Saturday. What was the median daily high over those six days? Give the number.
Show a hint
- Read the height of every bar first, all six of them, then write the temperatures in order from lowest to highest.
- With six values there is no single middle bar. Find the two temperatures sitting in the middle of your ordered list and average them.
Show the full solution
Sorted, the six highs are \(12, 15, 16, 18, 20, 22\). An even count has no single middle value, so average the 3rd and 4th, $$\frac{16+18}{2}=\frac{34}{2}=\boxed{17}$$ Bars sit in whatever order the categories come in, so sort the heights before looking for a median.
Practice these ideas
Practice
The table above shows the points each team scored in two rounds. Add up every entry to find the grand total of all points scored across both rounds. Give the number.
Show the solution
Total each round column, then combine. Round 1 gives \(30+41+28=99\) and Round 2 gives \(22+19+35=76\), so $$99+76=\boxed{175}.$$
Practice
The bar graph above shows how many votes each music genre received in a class poll. How many votes did rock get? Give the number.
Show the solution
The top of the rock bar lines up with the 18 mark on the vote scale, so rock got \(\boxed{18}\) votes.
Practice
The bar graph above shows how many new members the club gained each quarter. How many new members did the club gain over the whole year? Give the number.
Show the solution
The whole year is all four quarters together, so add the four bar heights, $$9+13+7+11=\boxed{40}.$$
Practice
The line graph above tracks a seedling's height in centimeters over six weeks. How tall was the seedling at week 4? Give the number.
Show the solution
Find week 4 on the horizontal axis and read its plotted point straight across to the height axis, which lands on \(\boxed{14}\) centimeters.
Practice
The line graph above tracks two runners' weekly miles over five weeks. On how many of the five weeks did Runner A run more miles than Runner B? Give the number.
Show the solution
Runner A's point sits above Runner B's in week 2, week 3, and week 5, and B is higher or tied in the other two weeks. That is \(\boxed{3}\) weeks. Two lines have to be compared point by point, one week at a time, rather than judged by their overall shape.
Practice
The pie chart above shows how 300 students get to school. How many of them take the bus? Give the number.
Show the solution
The bus slice is \(45\%\), so take 45 percent of the 300 students, $$0.45 \times 300 = \boxed{135}.$$ A percent only becomes a head count once you multiply it by the total.
Practice
The pie chart above shows how the 300 students get to school. How many students either walk or bike? Add those two slices, then find that percent of 300. Give the number.
Show the solution
Walk is \(20\%\) and bike is \(10\%\), so together they cover \(30\%\) of the students. Take that share of 300, $$0.30 \times 300 = \boxed{90}.$$
Practice
The pie chart above shows the favorite colors in a class. One slice, orange, is marked with a question mark instead of its percent. What percent of the class picked orange? What percent?
Show the solution
The four labeled slices add to \(28+22+19+16=85\), and the whole pie is 100 percent, so orange is what is left over, $$100 - 85 = \boxed{15}.$$
Practice
The stem-and-leaf plot above shows the eight scores from a bowling match. What is the median of these scores? Give the number.
Show the solution
The leaves already run smallest to largest, and with eight scores the median is the average of the 4th and 5th. Those two are 31 and 33, so $$\frac{31+33}{2}=\boxed{32}.$$
Practice
The stem-and-leaf plot above shows ten step-counts, in hundreds. How many days had 40 hundred steps or more? How many days?
Show the solution
Values of 40 hundred or more sit on stem 4 and stem 5. Stem 4 carries three leaves and stem 5 carries one, so the count is $$3 + 1 = \boxed{4}.$$ Once a cutoff lines up with a stem, you can count whole rows instead of reading every value.
Practice
The histogram above shows how many students have each number of siblings. What is the most common number of siblings among these students? Give the number.
Show the solution
The tallest bar sits over 2 siblings, so more students have 2 siblings than any other number, giving \(\boxed{2}\). The answer here is the label under the tallest bar, not the height of that bar.
Practice
The histogram above shows how many books each student read last month. How many students were surveyed in all? Give the number.
Show the solution
Each student lands in exactly one bar, so add the bar heights, $$3+7+8+5+1=\boxed{24}.$$ The heights count students, so summing them counts everyone surveyed once.
Practice
The pie chart above shows how 500 smoothie orders split across four flavors. How many more mango smoothies were ordered than kiwi? Give the number.
Show the solution
Mango is \(40\%\) of 500, so \(0.40 \times 500 = 200\) orders, and kiwi is \(15\%\) of 500, so \(0.15 \times 500 = 75\). Subtracting, $$200 - 75 = \boxed{125}.$$