S1 Mathematics Practice Book · Unit 7: Statistics with ungrouped data

Unit 7 Statistics with ungrouped data

Key unit competence: Organize real-world data using appropriate statistical methods and graphical representations.

Key ideas

Statistics is the collection, organisation, presentation, analysis and interpretation of data.
Qualitative data are categories (colour, district); quantitative data are numbers. Quantitative data are discrete (counted: number of pupils) or continuous (measured: height, mass).
A frequency table shows how often each value occurs; the cumulative frequency is the running total of frequencies.
Mean $\overset{x}{ˉ} = \frac{\sum x}{n}$ , or from a frequency table $\overset{x}{ˉ} = \frac{\sum f x}{\sum f}$ . Mode: the most frequent value. Median: the middle value of the ordered data (for $n$ values it is in position $\frac{n + 1}{2}$ ).
Range = largest − smallest. Quartiles $Q_{1}$ , $Q_{2}$ (median), $Q_{3}$ split ordered data into four equal parts; interquartile range = $Q_{3} - Q_{1}$ ; midrange $= \frac{min + max}{2}$ .
Graphs: pictogram, bar chart, pie chart (sector angle $= \frac{f}{\sum f} \times 36 0^{\circ}$ ), line graph, and cumulative frequency curve (ogive).

Formulas to know

\overset{x}{ˉ} = \frac{\sum x}{n}, \overset{x}{ˉ} = \frac{\sum f x}{\sum f}

median position = \frac{n + 1}{2}

sector angle = \frac{f}{\sum f} \times 36 0^{\circ}

I QR = Q_{3} - Q_{1}

Common misconceptions

"The median of 7, 2, 9, 4, 5 is 9 (the middle number as written)." No: order the data first: 2, 4, 5, 7, 9 so the median is 5.
"The mode is the highest frequency." No: the mode is the value that has the highest frequency.
"Mean from a frequency table = $\frac{\sum x}{number of rows}$ ." No: use $\frac{\sum f x}{\sum f}$ .

Worked examples

Example 1. The marks of 5 students are 12, 15, 11, 15, 17. Find the mean, mode, median and range.

Solution

Mean $= \frac{70}{5} = 14$ . Mode $= 15$ . Ordered: 11, 12, 15, 15, 17, so median $= 15$ . Range $= 17 - 11 = 6$ .

Example 2. In a class, 12 pupils walk, 8 cycle and 4 take a bus. Find the pie chart angle for each.

Solution

Total 24. Walk $\frac{12}{24} \times 36 0^{\circ} = 18 0^{\circ}$ , cycle $12 0^{\circ}$ , bus $6 0^{\circ}$ .

Practice questions

Easier and harder questions are mixed, just like in a real exam.

Classify as qualitative, discrete or continuous: (a) district of birth, (b) number of cows, (c) time to run 100 m.
Show answer
(a) qualitative, (b) discrete, (c) continuous.
Common mistake: Calling counted data continuous.
From a bar chart: Monday 20, Tuesday 35, Wednesday 25, Thursday 40 visitors. Convert it to a frequency table and find the mean number of visitors per day.
Show answer
Table: Mon 20, Tue 35, Wed 25, Thu 40. Mean $= \frac{120}{4} = 30$ visitors per day.
Common mistake: Dividing by the largest frequency.
The ages of 10 students are 13, 14, 13, 15, 14, 13, 16, 14, 13, 15. Make a frequency table with cumulative frequency.
Show answer
13: f=4, cf=4; 14: f=3, cf=7; 15: f=2, cf=9; 16: f=1, cf=10.
Common mistake: Forgetting that the last cumulative frequency must equal 10.
Daily rainfall (mm) for a week: 0, 4, 12, 0, 7, 3, 9. Draw a line graph and state which day had the most rain.
Show answer
Plot day (x-axis) against rainfall (y-axis) and join the points. Day 3 had the most rain (12 mm).
Common mistake: Not starting the y-axis at 0 or using unequal scales.
Using the table above, find the mean, mode and median age.
Show answer
Mean $= \frac{13 \times 4 + 14 \times 3 + 15 \times 2 + 16}{10} = \frac{140}{10} = 14$ . Mode 13. Median: 5th and 6th values are 14 and 14, so 14.
Common mistake: Using the frequencies as the data values.
The mean of 6 numbers is 9. When a seventh number is added, the mean becomes 10. Find the seventh number.
Show answer
$70 - 54 = 16$ .
Common mistake: Answering 10.
Find $Q_{1}$ , $Q_{3}$ and the IQR of 4, 6, 7, 9, 11, 12, 15.
Show answer
$Q_{1}$ is the 2nd value: 6; $Q_{3}$ is the 6th value: 12; IQR $= 6$ .
Common mistake: Not ordering the data first.
A family spends 50 000 FRW: food 25 000, transport 10 000, school 12 500, other 2 500. Find the pie chart angles.
Show answer
Food $18 0^{\circ}$ , transport $7 2^{\circ}$ , school $9 0^{\circ}$ , other $1 8^{\circ}$ (total $36 0^{\circ}$ ).
Common mistake: Using 100 instead of 360.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. Values 1, 2, 3 have frequencies 2, 5, 3. Find the mean.	a. 13
2. The mean of 5 numbers is 12. Four of them are 10, 15, 8 and 14. Find the fifth.	b. 2.1
3. Values 1, 2, 3 have frequencies 2, 5, 3. Find $\sum f x$ .	c. 21
4. Find the range of 12, 4, 19, 7, 15.	d. 15

Show answer

1 → b (2.1) · 2 → a (13) · 3 → c (21) · 4 → d (15)

$\frac{\sum f x}{\sum f} = \frac{21}{10} = 2.1$ .
Step 1. Total of the 5 numbers $= 5 \times 12 = 60$ .
Step 2. Fifth $= 60 - (10 + 15 + 8 + 14) = 60 - 47 = 13$ .
$1 \times 2 + 2 \times 5 + 3 \times 3 = 2 + 10 + 9 = 21$ .
$19 - 4 = 15$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
3 · an ogive · 5 · $Q_{3} - Q_{1}$
1. The ordered data 1, 3, 5, 7, 9, 11, 13 has lower quartile .
2. The interquartile range is .
3. A cumulative frequency curve is also called .
Show answer
1: 3 · 2: $Q_{3} - Q_{1}$ · 3: an ogive
- Position $\frac{n + 1}{4} = \frac{8}{4} = 2$ : the 2nd value is 3.
- It measures the spread of the middle half of the data.
- The ogive is the cumulative frequency curve.

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. Find the mode of 3, 5, 5, 6, 7, 5, 3.	a. $9 0^{\circ}$
2. Find the mean of 4, 7, 9, 10.	b. 35
3. In a pie chart of 60 people, 15 chose tea. What is the angle for tea?	c. 7.5
4. In a pictogram one symbol stands for 10 bags. How many bags do 3.5 symbols show?	d. 5

Show answer

1 → d (5) · 2 → c (7.5) · 3 → a ( $9 0^{\circ}$ ) · 4 → b (35)

5 occurs three times, more than any other value.
$\frac{30}{4} = 7.5$ .
$\frac{15}{60} \times 36 0^{\circ} = 9 0^{\circ}$ .
$3.5 \times 10 = 35$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
5 · 9 · 62 · 10
1. The mean mark of 10 students is 60. An eleventh student scores 82. Find the new mean. →
2. The mean of 6, 8, $x$ and 12 is 9. Find $x$ . →
3. Find the median of 7, 2, 9, 4, 5. →
Show answer
1: 62 · 2: 10 · 3: 5
- Step 1. Old total $= 10 \times 60 = 600$ ; new total $= 600 + 82 = 682$ .
  Step 2. New mean $= \frac{682}{11} = 62$ .
- Step 1. Total $= 4 \times 9 = 36$ .
  Step 2. $x = 36 - (6 + 8 + 12) = 36 - 26 = 10$ .
- Ordered: 2, 4, 5, 7, 9. The middle value is 5.

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