IP Maths Notes (Lower Sec, Year 1-2): 09) Data Handling & Statistics

Statistics converts raw data into decisions. Learn how to summarise, display, and interpret data sets with precision.

Learning targets

• Build frequency tables, cumulative frequency tables, and histograms.
• Compute mean, median, mode, and range (including grouped data estimates).
• Draw and interpret box plots and scatter plots.
• Write commentary sentences that connect statistics to context.

1. Organising data

1.1 Frequency tables

Example: Scores out of 10 — 5, 7, 8, 5, 6, 9, 7.

1.2 Cumulative frequency

Add frequencies progressively to track the number of observations below a threshold. Useful for median estimates in grouped data.

2. Measures of central tendency

• Mean: \( \bar{x} = \frac{\sum x}{n} \).
• Median: Middle value after sorting; if even number of data, average the two middle values.
• Mode: Most frequent value.

Worked example: Data set: 12, 15, 18, 19, 20, 20, 21, 25.

\[ \bar{x} = \frac{12 + 15 + 18 + 19 + 20 + 20 + 21 + 25}{8} = \frac{150}{8} = 18.75. \]

Median = average of 4th and 5th values → \( \frac{19 + 20}{2} = 19.5 \).

Mode = 20.

Range = 25 - 12 = 13.

3. Grouped data estimates

Estimate mean using midpoints \( x_i \) and frequencies \( f_i \):

\[ \bar{x} \approx \frac{\sum f_i x_i}{\sum f_i}. \]

Median class: locate where cumulative frequency crosses \( \frac{n}{2} \).

4. Data representations

• Bar chart: discrete categories.
• Histogram: continuous data (no gaps between bars).
• Pie chart: proportionate sectors (rare for IP exams, but know how to compute angles: \( \text{angle} = \frac{\text{frequency}}{n} \times 360^\circ \)).
• Box-and-whisker plot: shows median, quartiles, and spread.
• Scatter plot: pairs of values; look for correlation.

Worked example — Box plot interpretation

Five-number summary: minimum 28, lower quartile 34, median 41, upper quartile 47, maximum 55.

• Interquartile range (IQR) = 47 - 34 = 13 (measure of middle spread).
• 50% of data lies between 34 and 47.
• If comparing classes, comment on centres (median) and spreads (IQR).

5. Writing statistical commentary

Answer three questions:

1. What is the key statistic?
2. So what does it mean in context?
3. Now what: what action or conclusion follows?

Example sentence: “Class A's median score is 41, 3 marks higher than Class B's, suggesting stronger overall performance despite similar spreads.”

Try it yourself

1. A data set has values 4, 7, 7, 8, 9, 10, 10, 13. Compute mean, median, mode, and range.
2. Construct a frequency table and histogram for the following heights (cm): 150, 152, 154, 154, 155, 156, 157, 158, 158, 159, 160.
3. Given grouped data classes 0-10, 10-20, 20-30 with frequencies 6, 11, 5, estimate the mean.
4. Create a box plot for the data set {12, 18, 19, 23, 24, 25, 27, 31, 34} and interpret the median and IQR.

Complete the sequence with probability at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-10-Probability-Models.