Dot diagrams and stem-and-leaf diagrams questions with answers

Questions

1. Question 1Dot diagram: the mode

   The dot diagram shows the scores of pupils in a 10-mark quiz. Find the mode.

   Show the worked solution

   1. Step 1: Find the tallest column of dotsThe tallest column has $5$ dots.
   2. Step 2: The mode is the value under the tallest columnMode $= 7$
2. Question 2Dot diagram: the median

   The dot diagram shows the number of siblings of each pupil in a class. Find the median.

   Show the worked solution

   1. Step 1: Count the dots: each dot is one valueNumber of values $= 13$
   2. Step 2: The dots are already in order: find the position of the medianThe median is the $\dfrac{13 + 1}{2} = 7$th value.Median $= 1$
3. Question 3Stem-and-leaf: the range

   The stem-and-leaf diagram shows the masses of some parcels. Find the range.

   Show the worked solution

   1. Step 1: Read the largest and smallest values: use the key (stem = tens, leaf = ones)Largest value $= 42$ kg, smallest value $= 13$ kg
   2. Step 2: Range = largest − smallestRange $= 42 - 13$$= 29$ kg
4. Question 4Stem-and-leaf: the median

   The stem-and-leaf diagram shows the times taken by runners to complete a 5 km race. Find the median.

   Show the worked solution

   1. Step 1: Count the leaves: each leaf is one valueNumber of values $= 19$
   2. Step 2: The leaves are in order: find the position of the medianThe median is the $\dfrac{19 + 1}{2} = 10$th value.Median $= 37$ minutes
5. Question 5Back-to-back stem-and-leaf: one group

   The back-to-back stem-and-leaf diagram shows the heights, in cm, of seedlings grown with fertiliser X and with fertiliser Y. Find the range for fertiliser Y.

   Show the worked solution

   1. Step 1: Read group Y only: its leaves are on the right of the stemsGroup Y has $11$ values: $27,\ 32,\ 34,\ 40,\ 43,\ 44,\ 44,\ 48,\ 51,\ 53,\ 60$
   2. Step 2: Range = largest − smallestRange $= 60 - 27$$= 33$ cm
6. Question 6Back-to-back: compare the two groups

   The back-to-back stem-and-leaf diagram shows the times, in seconds, taken by some boys and girls to swim 50 m. By comparing the medians, state which group was faster.

   Show the worked solution

   1. Step 1: Find each group's median (count to the middle of each side)Median of Boys $= 57$ (12 values)Median of Girls $= 51$ (15 values)
   2. Step 2: A lower time means the group was fasterGroup Girls has the lower median, so it was faster.
7. Question 7An unknown leaf from the median or range

   The stem-and-leaf diagram shows the times taken by runners to complete a 5 km race. One leaf is shown as $x$. The median is $44$ minutes. Find the value of $x$.

   Show the worked solution

   1. Step 1: Count the leaves (x is one of them): where is the median?There are $11$ values.The median is the $6$th value: the leaf $x$, so the value $40 + x$.
   2. Step 2: Write an equation in x and solve$40 + x = 44$$x = 4$
8. Question 8Dot diagram: the mean or a percentage

   The dot diagram shows the shoe sizes of the members of a running club. Find the percentage of the values that are at least $39$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Count the dots at 39 and above (at least 39 includes 39)$12$ dots are at $39$ or above.There are $21$ dots in all.
   2. Step 2: Percentage = number ÷ total × 100%$\dfrac{12}{21} \times 100\%$$= 57.1428\ldots$$= 57.1$ % (3 s.f.)