Practice Maths

Measures of Centre and Spread Using Technology

Key Ideas

Key Terms

mean
sum of all values ÷ number of values. Affected by outliers.
median
middle value when data is ordered. For an even number of values, average the two middle values. Resistant to outliers.
mode
the most frequently occurring value. There may be more than one mode, or none.
range
maximum − minimum. A simple measure of spread.
quartiles (Q1, Q3)
Q1 (lower quartile) = median of the lower half; Q3 (upper quartile) = median of the upper half.
interquartile range (IQR)
IQR = Q3 − Q1. Measures the spread of the middle 50% of data. Not affected by outliers.
outlier
a value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
Hot Tip IQR = Q3 − Q1. It measures the spread of the middle 50% of data. Unlike range, IQR is not affected by outliers, making it a more reliable measure of spread for skewed data.

Worked Example

Dataset: {3, 5, 7, 9, 11, 13, 15}

Mean: (3+5+7+9+11+13+15) ÷ 7 = 63 ÷ 7 = 9

Median: 9 (the 4th value)

Q1: Median of lower half {3,5,7} = 5

Q3: Median of upper half {11,13,15} = 13

IQR: 13 − 5 = 8

Range: 15 − 3 = 12

Why Use Technology?

Calculating the mean, median, and range by hand works fine for small data sets. But in real life — and in exams — you will often work with larger data sets where calculating by hand would take too long or introduce errors. Scientific calculators and spreadsheet programs like Excel or Google Sheets can calculate these statistics instantly and accurately once you enter the data correctly.

Learning to use technology for statistics is not about avoiding the maths — it is about working efficiently and having more time to interpret and explain your results. You still need to understand what each statistic means and when to use it.

Using a Scientific Calculator for Statistics

Most scientific calculators (including the Casio fx-82AU PLUS, which is allowed in Queensland exams) have a statistics mode. The steps are:

  1. Press MODE and select STAT (or SD on older models).
  2. Choose 1-VAR (one variable statistics).
  3. Enter your data values, pressing = after each one.
  4. Press AC, then access the statistics results (usually via the SHIFT and STAT keys).
  5. From the menu, you can find: mean (x̄), sum (Σx), number of values (n), and more.

The calculator does not give median or mode directly — you still need to order the data and identify these yourself, or use a spreadsheet.

Using a Spreadsheet (Excel or Google Sheets)

In a spreadsheet, enter your data values in a column (e.g. A1 to A20). Then use these functions in any empty cell:

  • =AVERAGE(A1:A20) — calculates the mean.
  • =MEDIAN(A1:A20) — calculates the median.
  • =MODE(A1:A20) — calculates the mode (the most frequent value).
  • =MAX(A1:A20) - MIN(A1:A20) — calculates the range.
  • =QUARTILE(A1:A20, 3) - QUARTILE(A1:A20, 1) — calculates the interquartile range (IQR).

The IQR (interquartile range) is the range of the middle 50% of the data. It equals Q3 − Q1, where Q1 is the lower quartile (the median of the bottom half) and Q3 is the upper quartile (the median of the top half). The IQR is a better measure of spread than the range when there are outliers, because outliers do not affect it.

Interpreting Technology Output

Once technology gives you statistics, you need to interpret them in context. The mean is affected by extreme values (outliers) — one very high or very low value pulls the mean toward it. The median is resistant to outliers, making it better for describing the "typical" value when the data is skewed. Use the mean when the data is reasonably symmetric and the median when there are outliers or when the data is skewed.

The mode is most useful for categorical data (like "most popular colour") or when you specifically want to know the most common value. The range gives a quick sense of how spread out the data is, but it only uses the two extreme values. The IQR is more robust because it describes the spread of the middle 50%.

When to Use Which Measure

A handy guide: if you are reporting on income or house prices (which are often heavily skewed with a few very high values), use the median. If you are reporting on scores on a fair test or heights of students (roughly symmetric), use the mean. Always report the spread (range or IQR) alongside the centre, because knowing the average without knowing how spread out the data is gives an incomplete picture.

Key tip: Technology can make errors if you enter data incorrectly. Always do a quick sanity check: Does the mean look approximately in the middle of your data? Is the median close to the mean (if so, the data is roughly symmetric)? If the technology output looks wildly different from what you expect, recheck your data entry before assuming the answer is correct.

Mastery Practice

  1. Calculate the mean, median, mode, and range for each dataset. Fluency

    1. {4, 7, 9, 9, 11, 14}
    2. {3, 5, 5, 8, 10, 12, 12, 15}
    3. {20, 25, 25, 30, 35, 40, 40, 40, 50}
    4. {6, 8, 10, 10, 12, 14, 16, 20}
    5. {15, 18, 21, 21, 24, 27}
    6. {2, 4, 4, 6, 8, 8, 10, 12}
    7. {50, 55, 60, 65, 70, 70, 80}
    8. {100, 105, 110, 110, 115, 120, 120, 125, 130}

  2. Find Q1, Q3, and the IQR for each dataset. Order the data first. Fluency

    1. {8, 12, 14, 18, 22, 26, 30}
    2. {5, 7, 9, 11, 13, 15, 17, 19}
    3. {3, 6, 9, 12, 15, 18, 21, 24}
    4. {10, 14, 16, 20, 24, 28, 32, 36}
    5. {42, 45, 48, 51, 54, 57, 60}
    6. {7, 11, 13, 17, 19, 23, 25, 29, 31}
    7. {2, 5, 8, 11, 14, 17, 20, 23}
    8. {30, 35, 40, 45, 50, 55, 60, 65, 70}

  3. Use the rule: a value is a potential outlier if it is less than Q1 − 1.5 × IQR or greater than Q3 + 1.5 × IQR. Identify any outliers. Fluency

    1. {4, 6, 8, 10, 12, 14, 32} — Q1 = 6, Q3 = 14, IQR = 8
    2. {15, 17, 18, 20, 22, 24, 3} — Q1 = 16, Q3 = 23, IQR = 7
    3. {50, 52, 55, 58, 60, 62, 65, 95} — Q1 = 52, Q3 = 63.5, IQR = 11.5
    4. {30, 32, 34, 36, 38, 40, 42, 44, 1} — Q1 = 32, Q3 = 42, IQR = 10
    5. {10, 12, 12, 14, 15, 16, 18, 50} — Q1 = 12, Q3 = 17, IQR = 5
    6. {200, 210, 215, 220, 225, 230, 260, 500} — Q1 = 212.5, Q3 = 227.5, IQR = 15

  4. Compare the two datasets using mean, median, and IQR. Describe what the statistics tell you about each group. Understanding

    1. Class A test scores: {55, 60, 65, 70, 75, 80, 85}. Class B test scores: {40, 55, 65, 70, 75, 85, 100}. Which class has more spread in scores?
    2. Athlete A sprint times (seconds): {12.1, 12.3, 12.3, 12.4, 12.6, 12.8}. Athlete B sprint times: {11.8, 12.0, 12.4, 12.5, 12.9, 13.2}. Which athlete is more consistent?
    3. Plant heights (cm) with water: {18, 20, 22, 24, 26, 28}. Without water: {10, 12, 16, 18, 22, 24}. What do the statistics show about the effect of watering?
    4. Daily temperature (max, °C) in July: {14, 15, 17, 17, 18, 20, 22}. In January: {29, 31, 32, 33, 34, 35, 38}. Compare the centre and spread of temperatures in each month.
    5. Boys’ heights (cm): {155, 158, 161, 163, 166, 170, 172}. Girls’ heights: {152, 155, 158, 160, 163, 165, 168}. Which group has the greater IQR?
    6. House prices (in $000s) in Suburb A: {450, 480, 500, 510, 520, 540}. Suburb B: {380, 420, 510, 515, 520, 620}. Compare the median and IQR of each suburb.

  5. Write the spreadsheet formula you would use to calculate each measure, assuming data is in cells A1 to A20. Understanding

    1. Mean
    2. Median
    3. Mode
    4. Minimum value
    5. Maximum value
    6. Range (maximum minus minimum)
    7. Q1 (lower quartile)
    8. Q3 (upper quartile)

  6. Solve each extended problem using measures of centre and spread. Problem Solving

    1. A class of 10 students scored the following marks on a maths test: {58, 62, 65, 68, 70, 72, 75, 78, 80, 92}.
      (i) Calculate the mean, median, and range.
      (ii) Find Q1, Q3, and IQR.
      (iii) Is 92 a potential outlier? Show working.
      (iv) If the outlier is removed, how does the mean change?
    2. Two basketball players recorded their points per game over a season:
      Player A: {8, 10, 12, 14, 16, 18, 20, 22}
      Player B: {5, 8, 14, 15, 15, 18, 22, 27}
      (i) Calculate the mean and median for each player.
      (ii) Calculate the IQR for each player.
      (iii) Which player is more consistent? Use statistics to justify.
    3. A scientist records the daily rainfall (mm) at two weather stations over 9 days:
      Station 1: {2, 4, 5, 6, 7, 8, 9, 10, 11}
      Station 2: {0, 1, 3, 5, 7, 9, 11, 13, 15}
      (i) Find the median and IQR for each station.
      (ii) Which station has more variable rainfall? Explain using the IQR.
      (iii) Write the spreadsheet formulas you would use to confirm these values.

  7. For each dataset or context, state which measure of centre (mean, median, or mode) is most appropriate and explain why. Understanding

    1. House prices: $450 000, $480 000, $495 000, $510 000, $520 000, $1 800 000.
    2. Shoe sizes sold in a shop: 7, 8, 8, 9, 9, 9, 10, 10, 11.
    3. Test scores for a class: {62, 65, 68, 70, 71, 72, 74, 75, 77, 78}.
    4. Daily temperatures over a week: {18, 20, 22, 21, 23, 25, 24}.
    5. Number of goals scored per match: {0, 0, 1, 1, 1, 2, 2, 3, 8}.
    6. Favourite colours of students surveyed: red, blue, blue, green, blue, red, yellow.

  8. The original dataset is {10, 14, 16, 18, 20, 22}. A new value is added. Calculate the new mean and median for each case and compare to the original. Understanding

    Original mean = 16.67 (to 2 d.p.), original median = 17.

    1. A value of 16 is added.
    2. A value of 2 is added.
    3. A value of 50 is added.
    4. Which added value caused the biggest change in the mean? In the median?
    5. Which measure (mean or median) was least affected by adding the value 50? Why?

  9. Use the statistics given to answer each question. Understanding

    1. Two sprinters have identical mean times of 11.4 seconds. Sprinter A has IQR = 0.2 s; Sprinter B has IQR = 1.1 s. Who is more consistent? Explain.
    2. A dataset has Q1 = 30, Q3 = 50. Calculate the IQR and state what it tells you about the middle 50% of the data.
    3. Dataset A has IQR = 5; Dataset B has IQR = 25. If both have the same median, which dataset has more variable data?
    4. A dataset has Q1 = 12, Q3 = 20, IQR = 8. Is the value 35 a potential outlier? Show your working.
    5. The IQR for a dataset of student wait times is 6 minutes. What does this mean in context?

  10. Full investigation using measures of centre and spread. Problem Solving

    The following are the weekly pocket money amounts (in dollars) for two groups of students.

    Group A: 5, 8, 10, 10, 12, 15, 15, 18, 20, 25

    Group B: 2, 5, 8, 10, 12, 12, 15, 20, 30, 50

    1. Calculate the mean, median, and range for each group.
    2. Find Q1, Q3, and IQR for each group.
    3. Check whether any values in Group B are potential outliers.
    4. Which group has more spread in their pocket money? Use statistics to support your answer.
    5. Which measure of centre (mean or median) better represents the typical amount for Group B? Justify using the outlier check.
    6. Write the spreadsheet formulas (=AVERAGE, =MEDIAN, =QUARTILE) you would use to verify your answers for Group A in cells A1:A10.
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