Statistics — Student Revision Notes

Junior Cycle Mathematics • Statistics learning outcomes

1. Asking good statistical questions

A statistical question expects variability and refers to a group (not a single case).

  • ✔ “What is the typical journey time to school for 2nd years?”
  • ✘ “What is my journey time today?”
Checklist: anticipate variability • identify the population • choose a measurable variable

Quick practice

  1. Rewrite “How tall am I?” as a statistical question.
  2. Identify the population and variable in: “What is the average number of apps on students’ phones?”

Ans 1. “What are the heights of students in 2nd year?” (or “What is the median height of 2nd years?”)

Ans 2. Population: the specified student group. Variable: number of apps on a student’s phone (discrete numerical).

2. Collecting and organising data

Use fair methods (avoid bias), then record results in a frequency table.

Frequency Distribution Table

Values
4
5
6
7
8
9
Frequencies
2
4
7
5
3
1

Two rows: first the values, then the matching frequencies.

Grouped Frequency Distribution Table

Class intervals (cm)
140–149
150–159
160–169
170–179
180–189
Frequencies
2
6
9
7
1

Equal-width classes; boundaries determined by context.

Quick practice

  1. Give one way sampling bias could occur in a phone-use survey, and how to reduce it.
  2. From the shoe-size table, how many students are size 6 or larger?

Ans 1. Surveying only near the computer room at lunch over-represents tech-keen students. Reduce by random sampling across classes/times.

Ans 2. Size \(6+\): \(7+5+3+1=16\) students.

3. Types of data

Categorical (labels, e.g. eye colour), Numerical — discrete (counts, e.g. siblings), Numerical — continuous (measurements, e.g. height).

Quick practice

  1. Classify: “hours of sleep last night” and “bus route taken”.

Ans. Hours of sleep: numerical continuous. Bus route taken: categorical (nominal).

4. Graphical displays

Choose the display to suit the data. Always include a title, labels, and sensible scales.

Bar chart (categorical/discrete)

Bars are equal width and do not touch.

Pie chart (parts of a whole)

Angles proportional to the counts/percentages.

Histogram (continuous, equal intervals)

Bars touch. Frequency shown by area/height.

Stem-and-leaf plot (numerical data)

Key: 6 | 4 means 64. Shows the raw values and the distribution.

Quick practice

  1. From the histogram, which class contains the modal height?
  2. Give one way a bar chart could mislead a reader.
  3. From the stem-and-leaf, state the median score.

Ans 1. The class with frequency \(9\): \(160\!-\!169\) cm.

Ans 2. Truncated vertical axis (not starting at 0), uneven bar widths, or missing labels.

Ans 3. Median = [will appear once data render].

5. Averages (measures of central tendency)

\(\displaystyle \bar{x}=\frac{\sum x_i}{n},\quad \text{Median}=\text{middle value (ordered)},\quad \text{Mode}=\text{most frequent}\).

For grouped data, use class mid-interval values to estimate the mean.

Quick practice

  1. Data: \(7,8,8,12,14\). Find the mean, median, and mode.

\(\sum x=49\Rightarrow \bar{x}=49/5=9.8\). Ordered \(7,8,8,12,14\): median \(=8\). Mode \(=8\).

6. Variability (spread)

Range \(=\text{highest}-\text{lowest}\). A small range means the data are tightly clustered.

Quick practice

  1. Two classes have the same mean test score. One has range \(=6\), the other \(=18\). Which class is more variable?

The class with range \(18\) is more variable (wider spread).

7. Estimate the mean from a grouped frequency table

Use the midpoint \(m\) of each class and multiply by its frequency \(f\). Then \(\displaystyle \bar{x}_{\text{est}}=\frac{\sum f m}{\sum f}\).

Class intervals (marks)
0–9
10–19
20–29
30–39
40–49
Frequencies \(f\)
3
6
10
7
4

Midpoints \(m=\{4.5,14.5,24.5,34.5,44.5\}\).

\(fm=\{13.5,87,245,241.5,178\}\).

\(\sum f=30\), \(\sum fm=765\Rightarrow \bar{x}_{\text{est}}=765/30=25.5\) marks.

This is an estimate because we represent each class by its midpoint.

Student task (use the repeated table below)

Use this grouped height table to estimate the mean height.

Class intervals (cm)
140–149
150–159
160–169
170–179
180–189
Frequencies \(f\)
2
6
9
7
1
  1. Compute the midpoints for each class and hence estimate \(\bar{x}\).
  2. Explain why this estimate may differ from the exact mean of the raw data.

Ans 1. Midpoints \(=\{144.5,154.5,164.5,174.5,184.5\}\). \(\sum f=25\).

\(\sum fm=144.5(2)+154.5(6)+164.5(9)+174.5(7)+184.5(1)=4109.5\Rightarrow \bar{x}_{\text{est}}=4109.5/25\approx164.38\ \text{cm}.\)

Ans 2. We approximated every value in a class by its midpoint; the real values within each interval are spread about that midpoint.

Statistics Tutorials

Data

Collecting Data

Mean, Mode and Median - Quartiles and Range

Presenting Data