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Statistics Exam Tips: Mean, Median, Mode, and IQR for O Level & IGCSE Maths

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Understanding Core Statistical Measures

  • Mean, median, mode and interquartile range (IQR) measure different aspects of data

  • Mean shows central value based on total sum

  • Median shows middle position

  • Mode shows most frequent value

  • IQR shows spread between Q₁ and Q₃

  • Always identify whether:

    • Data is raw

    • Grouped

    • Continuous

    • Discrete

  • Ensure correct handling based on data type

  • Choose appropriate measure for context

Mean: The Most Sensitive Measure of Central Tendency

  • Mean = sum of values ÷ number of values

  • Best measure when data has no outliers

  • Steps

    • Add all values

    • Divide by number of terms

  • Non-calculator tricks

    • Group numbers that add to easy totals

    • Use multiples of 10 strategy

    • Use balancing method for large datasets

  • Mean increases if a large outlier is added

  • Mean decreases if small outlier added

  • Mean often requires careful arithmetic in Paper 1

  • Check denominator before dividing

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Mean from Frequency Tables

  • Mean formula for frequency table

    • mean = (Σfx) ÷ (Σf)

  • Steps

    • Multiply each value by its frequency

    • Add all fx values

    • Add all frequencies

    • Divide Σfx by Σf

  • Used in grouped and ungrouped tables

  • Avoid errors

    • Forgetting to multiply first

    • Forgetting to sum correctly

    • Using incorrect Σf

Mean for Grouped Data (Midpoint Method)

  • Midpoint used when exact values unknown

    • midpoint = (lower boundary + upper boundary) ÷ 2

  • Mean = (Σf × midpoint) ÷ Σf

  • Recognise approximated mean

  • Examiners penalise using interval endpoints instead of midpoint

  • Grouped mean typically used when

    • Intervals like 10–20, 20–30 appear

    • Data continuous

  • Midpoint essential for accuracy

Median: The Middle Value

  • Median used when data contains extreme outliers

  • Median formula for ungrouped data

    • If n odd → middle value

    • If n even → average of two middle values

  • Arrange data in order (ascending or descending)

  • Avoid mixing values before sorting

  • Used for:

    • Box plots

    • Cumulative frequency graphs

    • Central tendency analysis

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Median from Frequency Tables

  • Find total frequency N

  • Identify median position

    • median position = (N + 1)/2

  • Cumulative frequency needed

  • Steps

    • Construct CF column

    • Locate interval where CF ≥ median position

    • That interval contains median

  • Exact median from tables sometimes computed using interpolation

Median from Cumulative Frequency Graphs

  • Draw ogive (cumulative frequency curve)

  • Locate N/2 on y-axis

  • Draw horizontal to curve

  • Drop vertical to x-axis

  • Read value accurately

  • Avoid drawing straight-line segments between unequal scales

  • Use fine pencil for precision

Mode: Most Frequent Value in the Data

  • Mode used for

    • Categories

    • Discrete repeated values

    • Quick identification

  • Advantages

    • Not affected by outliers

    • Simple to identify

  • Limitations

    • Not unique when two values appear equally

    • Not useful in continuous grouped data

Modal Class in Grouped Frequency Tables

  • Class with highest frequency is modal class

  • Exact mode not required

  • Useful for shape interpretation

  • Often tested with histogram comparison

  • Use frequency density for unequal class widths

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Interquartile Range (IQR): Spread of Middle 50 Percent

  • IQR = Q₃ − Q₁

  • Q₁ = 25th percentile

  • Q₃ = 75th percentile

  • Measures spread excluding extremes

  • Large IQR → data widely spread

  • Small IQR → data tightly clustered

Finding Quartiles from Data List

  • Arrange data in order

  • Find positions

    • Q₁ at N/4

    • Q₃ at 3N/4

  • Use midpoint method for even totals

  • Avoid confusion between position and value

  • IQR essential for comparing data sets

Quartiles from Frequency Tables

  • Need cumulative frequency

  • Q₁ = first value where CF ≥ N/4

  • Q₂ = first value where CF ≥ N/2

  • Q₃ = first value where CF ≥ 3N/4

  • Identify correct interval quickly

Quartiles from Cumulative Frequency Graph

  • Locate N/4 and 3N/4 on y-axis

  • Use horizontal → curve → vertical procedure

  • Read quartile values on x-axis

  • Draw neat and accurate lines

  • Used heavily in Paper 2/4 exam diagrams

Box-and-Whisker Plots (Boxplots)

  • Represent

    • minimum

    • Q₁

    • median

    • Q₃

    • maximum

  • IQR represented as length of box

  • Used for comparing two data distributions

  • Examiners reward

    • Correct box structure

    • Accurate whisker positions

    • Clear quartile labelling

Choosing Between Mean, Median and Mode

  • Use mean when

    • Data has no extreme outliers

    • Requires average with precision

  • Use median when

    • Outliers present

    • Skewed distributions

  • Use mode when

    • Categories

    • Shoe sizes

    • Most popular choice

  • Use IQR when

    • Spread required

    • Comparing variability

    • Boxplot interpretation

Impact of Outliers on Statistical Measures

  • Outlier = value very different from others

  • Outlier increases mean drastically

  • Outlier hardly affects median

  • Outlier does not affect mode

  • Outlier increases range

  • Outlier may increase IQR slightly

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

Mean vs Median vs IQR for Comparison Questions

  • To compare two data sets, always comment on:

    • Central tendency → mean or median

    • Spread → IQR or range

Example comparison statements

  • “Set A has a higher median, so values tend to be larger.”

  • “Set B has a smaller IQR, so values are more consistent.”

  • “Set A has a greater spread, indicating greater variability.”

  • “Set B has an outlier which affects the mean significantly.”

Frequency Table Exam Tricks

  • Always create CF column when dealing with median or IQR

  • Multiply systematically for Σfx

  • Check Σf carefully before calculating mean

  • Identify modal class for mode-related tasks

  • Use midpoints only for grouped mean

  • Avoid midpoint usage for median

  • Common trap:

    • Using midpoint to find mode or median (incorrect)

Cumulative Frequency Curve Interpretation

  • Increasing curve only

  • Steeper slope → high concentration of values

  • Flatter slope → spread-out data

  • Compare two curves for:

    • Median difference

    • Quartile difference

    • Overall spread

Histograms and Frequency Density

  • Used when class widths are unequal

  • Frequency density = frequency ÷ class width

  • Highest bar height ≠ highest frequency

  • Modal class from histogram

    • tallest bar = modal class

  • Used alongside CF graphs for full distribution analysis

Range vs IQR

  • Range = max − min

  • IQR = Q₃ − Q₁

  • Range strongly affected by outliers

  • IQR resistant to outliers

  • Use IQR when asked “compare spread properly”

  • Use range when data not continuous

Importance of Graph Accuracy

  • Ogive accuracy determines quartile accuracy

  • Incorrect plotting leads to wrong IQR

  • Draw smooth curve

  • Avoid line segments

  • Label axes clearly

  • Use consistent scale

  • Recheck plotted CF points

Word Problems in Statistics

  • Reading exam context carefully

  • Extract data correctly

  • Identify whether grouped or ungrouped

  • Determine which measure requested

  • Use mean for average

  • Use median for middle tendency

  • Use IQR for consistency measurement

  • Interpret comparisons with correct keywords

A Star Techniques for Statistics

  • Always build CF table when unsure

  • Always sort data before median

  • Always double-check Σfx and Σf

  • Use midpoint only for grouped mean

  • Use interpolation when allowed

  • Read quartiles precisely from graph

  • Comment with statistical reasoning

  • Avoid casual language like “bigger” or “smaller”

  • Use technical words

    • “Tends to be higher”

    • “Less variable”

    • “More consistent distribution”

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

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