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Data Handling and Statistics

Secondary 1 Mathematics · G3 focus · Study guide

Collect and organise data, choose and interpret charts, recognise misleading displays and calculate simple averages.

Concept → worked examples → practice20 min read · practice linked below
The same frequencies as a bar graph and pictogramThe bar graph has frequency ticks zero, four, eight and twelve; Bus has twelve students, Walk eight, Cycle four. Beside it, the pictogram shows three circles for Bus, two for Walk and one for Cycle. Its key states that one circle means four students.Number of students04812BusWalkCycleKey: ● = 4 studentsBusWalkCycleDifferent displays; the same 24 observations.

Start with a question and reliable data

Statistics helps us collect, organise and interpret information. First decide what you want to find out, whom or what you will observe, and how each observation will be recorded. A population is the whole group of interest; a sample is the part actually observed. A survey of one class tells you about that class; it may not represent the whole school.

Choose categories that do not overlap and that cover every possible response, or include an “other” category. Ask neutral questions: “Which journey do you usually make?” is clearer than “Don't you prefer the faster bus?” For measurements, state the unit and the rule for rounding. Record one observation per person unless multiple responses are intended.

Categorical data describe groups, such as transport type. Numerical data describe counts or measurements, such as number of siblings or journey time. A frequency is the number of times a category or value occurs. Tally marks help count; a frequency table makes the final counts easy to compare.

Worked example 1: a frequency table, bar graph and pictogram

Each of 24 students chooses their usual journey to school. There are 12 bus users, 8 walkers and 4 cyclists. Organise the results and find the fraction who walk.

Usual journey to school; one response per student
JourneyFrequency
Bus12
Walk8
Cycle4
Total24
  1. Check that every student is counted once: 12+8+4=2412+8+4=24.
  2. The fraction who walk is 824=13.\frac{8}{24}=\boxed{\frac13}.
  3. For a bar graph, label the categories, use equal-width separated bars, and show a numerical frequency scale starting at zero.
  4. For a pictogram with one symbol representing 4 students, draw 12÷4=312\div4=3 symbols for Bus, 2 for Walk and 1 for Cycle.
The same frequencies as a bar graph and pictogramThe bar graph has frequency ticks zero, four, eight and twelve; Bus has twelve students, Walk eight, Cycle four. Beside it, the pictogram shows three circles for Bus, two for Walk and one for Cycle. Its key states that one circle means four students.Number of students04812BusWalkCycleKey: ● = 4 studentsBusWalkCycleDifferent displays; the same 24 observations.
Read the pictogram key before counting. A half-symbol would represent 2 students with this key.

Check: six symbols represent 6×4=246\times4=24 students. The heights of the bars show frequencies, not the number of symbols in the pictogram.

Choose a display that answers the question

What each representation makes easy to see
DisplayUseful forWatch out for
TableReading precise values and organising several variablesA pattern may be less obvious than on a graph.
Bar graphComparing categories by frequencyUse equal bar widths and a consistent, normally zero-based scale.
PictogramShowing a small set of counts simplyThe symbol key matters; fractions of symbols must be clear. Changing icon size can exaggerate differences.
Line graphFollowing a quantity over an ordered sequence, often timeKeep time spacing proportional. Joining unrelated categories suggests a meaningless trend.
Pie chartShowing proportions of one wholeAll sectors together represent one total. Small differences and many categories are hard to compare.

A large sector in a small group's pie chart need not represent more people than a smaller sector in a much larger group. Compare the totals as well as the percentages.

Pie charts convert a fraction into an angle

The whole circle is 360∘360^\circ. A category occupying one quarter of the observations needs one quarter of that angle. Therefore:

sector angle=category frequencytotal frequency×360∘.\text{sector angle}=\frac{\text{category frequency}}{\text{total frequency}}\times360^\circ.category frequency=sector angle360∘×total frequency.\text{category frequency}=\frac{\text{sector angle}}{360^\circ}\times\text{total frequency}.

To draw a pie chart, calculate every angle, check their total is 360∘360^\circ, draw a circle and a starting radius, then measure successive sectors with a protractor. Label the categories and state the total; do not rely on colour alone.

Worked example 2: find an angle and recover a count

A club has 40 members: 18 choose badminton, 12 choose swimming and 10 choose running. Find the swimming sector angle. A different club of 60 members has a running sector of 84∘84^\circ; find its number of runners.

  1. Swimming makes up 12/4012/40 of the first club, so swimming angle=1240×360∘=108∘.\text{swimming angle}=\frac{12}{40}\times360^\circ=\boxed{108^\circ}.
  2. Check the other sectors: badminton =18/40×360=162∘=18/40\times360=162^\circ; running =10/40×360=90∘=10/40\times360=90^\circ. Their sum is 162+108+90=360∘162+108+90=360^\circ.
  3. For the second club use its total of 60: runners=84360×60=14.\text{runners}=\frac{84}{360}\times60=\boxed{14}.

Check: 14/60=7/3014/60=7/30 and (7/30)360=84(7/30)360=84. A sector angle is measured in degrees; the answer to “how many members?” is a whole-number count.

Turn counts into a pie chart

For 12 bus users, 8 walkers and 4 cyclists, the total is 24. The sectors are 180°, 120° and 60°. Changing a count changes the total, so it can change every sector's angle.

Travel choices shown as parts of a circleTwenty-four students: Bus twelve, fifty percent, one hundred eighty degrees; Walk eight, one third, one hundred twenty degrees; Cycle four, one sixth, sixty degrees. Labels state every count and angle.Each sector is a shareof the same whole.Bus: 12 → 180°Walk: 8 → 120°Cycle: 4 → 60°Total: 24 students
The text lists every category and angle. A zero count has no sector; with no observations at all, there is no pie to divide.

Total 24. Bus: 12/24 × 360° = 180°. Walk: 8/24 × 360° = 120°. Cycle: 4/24 × 360° = 60°. Angles total 360°.

Read a line graph using both axes

Read the horizontal coordinate first, then the vertical value and its unit. A line sloping upward means an increase; downward means a decrease; horizontal means no change. The steepness depends on both axis scales, so do not compare slopes across differently scaled charts by appearance alone.

Worked example 3: change over time, with a limit to the conclusion

The water level in a tank is recorded at 08:00, 09:00, 10:00 and 11:00 as 20, 32, 29 and 41 cm. Find the greatest hourly increase, the total change, and an estimate at 09:30 if the level changes steadily between readings.

Tank water level over three hoursHorizontal axis shows times eight, nine, ten and eleven a.m. Vertical axis is water level in centimetres from zero to fifty in equal ten-centimetre steps. Points are twenty, thirty-two, twenty-nine and forty-one centimetres respectively, joined by straight segments.Water level (cm)0102030405008:0009:0010:0011:0020322941Time
Straight joins help show the trend between recorded values; they do not prove what happened at every unrecorded instant.
  1. Subtract successive levels: 32−20=1232-20=12, 29−32=−329-32=-3, 41−29=1241-29=12 cm. The greatest hourly increase is 12 cm\boxed{12\text{ cm}}, during both 08:00–09:00 and 10:00–11:00.
  2. Total change is final minus initial: 41−20=21 cm41-20=\boxed{21\text{ cm}}. Check: 12−3+12=2112-3+12=21.
  3. 09:30 is halfway between 09:00 and 10:00. Under the stated steady-change assumption, (32+29)÷2=30.5 cm.(32+29)\div2=\boxed{30.5\text{ cm}}.

The 09:30 value is an estimate between observations. The graph alone does not tell us why the level fell or justify extending the same trend beyond 11:00.

Ask whether the display supports the claim

Check the source, sample, dates, units, axis starting values, tick spacing, pictogram key and totals. A graph can contain correct numbers yet encourage a misleading comparison. An association between two quantities does not by itself show that one causes the other.

Worked example 4: a bar can look three times as tall

A shop sells 40 tickets on Monday and 50 on Tuesday. A bar chart starts its vertical axis at 35 tickets. Someone claims that Tuesday's sales are three times Monday's. Assess the claim.

The same ticket counts with two different vertical baselinesLeft bar graph starts at zero; Monday forty tickets and Tuesday fifty tickets have heights in the ratio four to five. Right bar graph starts at thirty-five; the visible heights above that baseline are five and fifteen, in the ratio one to three. Both label the true counts forty and fifty. The truncated chart exaggerates their relative size.Baseline 0 ticketsBaseline 35 tickets050355040504050MonTueMonTueRead the counts and scale before comparing heights.
The right chart displays only the amount above 35. Its visible bar heights do not represent the full counts.
  1. Actual ratio of sales: 50÷40=1.2550\div40=1.25. Tuesday sold 1.25 times\boxed{1.25\text{ times}} Monday's count, not three times.
  2. The increase is 50−40=1050-40=10 tickets, which is 10/40×100=25%10/40\times100=\boxed{25\%}.
  3. The visible bar heights above 35 represent 40−35=540-35=5 and 50−35=1550-35=15 tickets. Their ratio is 15/5=315/5=3, explaining the misleading visual impression.

Conclusion: use a zero baseline for bar-length comparisons and show the scale clearly. A line graph may use a shortened scale to show small changes, but the axis and the conclusion still need careful reading.

Dot diagrams, grouped tables and averages

A dot diagram places one dot per observation above its numerical value. Stacked dots show repeated values. For a display with 1 dot above 0, 3 above 1 and 2 above 2, there are 1+3+2=61+3+2=6 observations; the total of their values is 0(1)+1(3)+2(2)=70(1)+1(3)+2(2)=7, not 6. Always check whether a dot represents one person, one group, or another unit.

A dot diagram shows six observationsHorizontal axis shows number of siblings zero, one and two. One dot is above zero, three dots above one and two dots above two. Each dot represents one student, so there are six students altogether.Key: one dot = one student012Number of siblings
Count dots for the number of students; read the horizontal value for how many siblings each student has.

A histogram shows numerical data grouped into intervals. For equal-width intervals, frequency can be plotted on the vertical axis, with touching bars to show adjacent intervals. A bar graph for separate categories has gaps. The modal class is the interval with the greatest frequency; it does not tell you an exact most common value inside that interval.

Worked example 5: place boundary values in the right class

Journey times, in minutes, are 6,10,12,17,20,20,24,28,30,356,10,12,17,20,20,24,28,30,35. Group them into 0<t≤100<t\le10, 10<t≤2010<t\le20, 20<t≤3020<t\le30, and 30<t≤4030<t\le40. Identify the modal class.

  1. The first class includes 10 but not 0, so it contains 6 and 10: frequency 2.
  2. The second includes 20 but not 10: 12, 17, 20, 20 give frequency 4. The third contains 24, 28, 30: frequency 3. The last contains 35: frequency 1.
  3. Check the frequencies: 2+4+3+1=102+4+3+1=10, matching the number of observations. The greatest frequency is 4, so the modal class is 10<t≤20\boxed{10<t\le20} minutes.
Histogram of journey times in equal ten-minute intervalsThe horizontal axis has class boundaries zero, ten, twenty, thirty and forty minutes. Touching equal-width bars have frequencies two, four, three and one. The vertical frequency scale runs from zero to four. The ten-to-twenty-minute class is the modal class.Frequency01234010203040Journey time (minutes)
The intervals all have width 10 minutes, so comparing heights compares frequencies. The inequalities in the question decide where exact boundary values belong.

Check: a time of 20 belongs to the second class once, never to both the second and third. Equal-width grouping loses the individual values, even though it makes the overall pattern clearer.

The mean is the total of all values divided by the number of values. The median is the middle value after sorting; for an even count, average the two middle values. The mode is a most frequent value; a data set can have no unique mode or more than one mode. The range is largest minus smallest.

Worked example 6: averages from a frequency table

Six students read these numbers of books in a month: 0,1,1,1,2,70,1,1,1,2,7. Find the mean, median, mode and range, and say which average describes a typical student best.

  1. The total number of books is 0+1+1+1+2+7=120+1+1+1+2+7=12, so the mean is 12÷6=212\div6=\boxed{2} books. Equivalently, use frequencies: 0(1)+1(3)+2(1)+7(1)1+3+1+1=2.\frac{0(1)+1(3)+2(1)+7(1)}{1+3+1+1}=2.
  2. The list is already sorted. Positions 3 and 4 are both 1, so the median is (1+1)÷2=1(1+1)\div2=\boxed{1} book.
  3. The most frequent value is 1, appearing three times, so the mode is 1\boxed{1} book. The range is 7−0=77-0=\boxed{7} books.
  4. The median of 1 is a useful description of the middle student here; the value 7 pulls the mean upward. The mean of 2 is useful if you want the equal share of the total books.

Check: mean × number of students =2×6=12=2\times6=12, the correct total. With a frequency table, divide by the total frequency, not by the number of distinct values. No one average is best for every question.

If only grouped intervals are known, the exact individual values and exact mean cannot usually be recovered. Replacing values in each interval by its midpoint gives an estimated mean, ∑(f×midpoint)/∑f\sum(f\times\text{midpoint})/\sum f; keep the word “estimated”.

Common mistakes and checks

  • Counting symbols without the key: multiply by the number each symbol represents.
  • Using the wrong total in a pie chart: identify the whole group before calculating a count or angle.
  • Confusing a value with its frequency: three students reading one book contribute 3 books in total.
  • Finding a median before sorting: sort first, then locate the middle position or positions.
  • Putting a boundary value into two intervals: read << and ≤\le carefully and count it once.
  • Claiming a cause from a trend: describe the observed association and consider other possible explanations.

Practise interpreting the evidence

Use our Sec 1 data handling and statistics question bank for tables, pie charts, line graphs, dot diagrams, histograms and averages. Read every axis and key before calculating, then finish with a sentence about the original data.