IP EMaths Notes (Upper Sec, Year 3-4): 15) Statistics and Data Handling

08 Nov 2025, 00:00 Z

Statistics questions reward tidy tables and well-labelled graphs. Document your calculator steps so you can replicate them under exam conditions.

Quick reference

Mean of grouped data: \( \bar{x} = frac{\sum fx}{\sum f} \).
Median for grouped data: locate \( n/2 \) on the cumulative frequency column and interpolate within that class.
Quartiles and percentiles: use cumulative frequency or ordered lists depending on the dataset size.
Variance/standard deviation (for raw data): with calculator support, record the key inputs in case you need to justify the button presses.
Correlation language: “positive”, “negative”, “no correlation”; do not say “strong cause”.
Box-and-whisker charts compare spread and medians; comment on both when writing conclusions.

List raw data in ascending order; tally if the dataset is long.
Build frequency and cumulative frequency columns side by side.
For grouped data, add a midpoint column and a \( fx \) column.
For two-variable data (x, y), include \( x^2 \), \( y^2 \), and \( xy \) columns if you need regression or product-moment correlation coefficient (PMCC) later.

Mean: use the \( \sum fx \) column. For raw data, type all values into the calculator’s statistics mode; state “1-VAR” on the calculator if required.
Median: for raw data, pick the middle value(s). For grouped data, locate \( n/2 \) within the cumulative frequencies and interpolate. For the example above, \( n = 40 \), so \( n/2 = 20 \) falls in the 40-60 class: median ≈ 44 minutes (linear interpolation).
Mode: from grouped data, quote the modal class. If the mode is needed more precisely, use the mode estimation formula (optional at IP level).

Range: highest minus lowest.
Interquartile range (IQR): \( Q_3 - Q_1 \); less sensitive to outliers than the range.
Variance / standard deviation: for ungrouped data, use calculator output (write down the x̄ and σ²). For grouped data, use the midpoints with the frequency column.

Using the table above:

Mean = \( frac{1860}{40} = 46.5 \) minutes.
Median: cumulative frequencies 6, 18, 28, 35, 40. The 20th observation sits in the 40-60 class; interpolating gives 44 minutes to the nearest whole minute.
Quartiles: \( Q_1 \) is the 10th observation (≈ 32 minutes), \( Q_3 \) is the 30th observation (≈ 61 minutes). Quote IQR ≈ 29 minutes.

Two classes recorded the time spent on revision (minutes per day). Box plots show:

Interpretation:

Class A has a higher median, so its typical student spends more time revising.
Class B has a much wider IQR, so revision times vary more; some students revise significantly more or less than the typical value.

Always comment on both location (median) and spread (IQR, range) when comparing box plots.

A set of ten students recorded their hours of supervised revision (x) and mock exam scores (y). The calculator outputs warning that r = 0.86.

State: There is a strong positive correlation between hours and mock scores.
Interpretation: More supervised revision tends to align with higher mock scores; however, correlation does not imply that supervision alone causes the improvement. Other factors (student motivation, resources) may contribute.
Use the line of best fit to make predictions only within the observed data range (interpolation). Extrapolation outside the range is risky.

The heights (cm) of 50 students are grouped in classes of width 5 cm. Outline the steps you would use to estimate the mean height and the median height.
A box plot for Class C shows median 55, lower quartile 40, upper quartile 75, minimum 25, maximum 90. Summarise what this tells you about the distribution of time spent on co-curricular activities per week.
Calculator output for paired data gives \( \bar{x} = 6.2 \), \( \bar{y} = 72 \), \( \sigma_x = 1.4 \), \( \sigma_y = 12 \), and PMCC r = 0.15. Comment on the strength of the linear relationship.