🇨🇩 Statistics: data handling

Tables, graphs and means

⏱ 25 minOfficial programme: CD-E7-STA-01

Your answers are checked, but your progress is not saved. To save it, sign in with your family code, or ask a parent to open this course from the dashboard.

🎯 In this lesson, you will learn to

Build frequency tables and graphs.
Compute simple and weighted means.

The lesson

Introduction

To prepare a parents’ meeting, the school in Isiro asks each 7th-grade pupil how many children live in their family. With the answers, a table and a chart must be made and a mean computed. This is the work of statistics: organising data so that it is easy to read. In this lesson you learn how to do it.

Explanation

Vocabulary and frequency table

The population is the whole set studied; each element is an individual.
The variable is what we observe. It is qualitative (colour, crop, means of transport) or quantitative (age, mass, number of children).
The frequency count of a value is the number of individuals with that value; the total count $N$ is the sum of the counts.
The relative frequency of a value is $\dfrac{\text{count}}{N}$ ; it is often given as a percentage. The relative frequencies add up to $1$ , that is $100\,\%$ .

A frequency table gives, for each value, its count and its relative frequency.

Explanation

Charts

Stick chart (quantitative variable): the values on the horizontal axis; for each value, a vertical bar whose height is the count.
Bar chart (qualitative variable): same idea, with separate bars for each category.
Pie chart: the disc stands for the whole population; each value gets a sector whose angle is proportional to its count:
$\text{angle}=\frac{\text{count}}{N}\times360^\circ$

Always give the chart a title and show the units on the axes.

Explanation

Means

Simple arithmetic mean of $n$ numbers: their sum divided by $n$ .
Weighted mean: when each value $x_i$ has a count (or a coefficient) $n_i$ ,
$\bar{x}=\frac{n_1x_1+n_2x_2+\dots+n_kx_k}{n_1+n_2+\dots+n_k}$

Divide by the sum of the counts (or of the coefficients), not by the number of different values.

Worked example

Problem: $20$ families were asked how many children they have. Results: $0$ children: $2$ families; $1$ : $4$ ; $2$ : $6$ ; $3$ : $5$ ; $4$ : $3$ . Make the frequency table and compute the mean number of children.

Step 1: Relative frequencies: $\dfrac{2}{20}=10\,\%$ ; $\dfrac{4}{20}=20\,\%$ ; $\dfrac{6}{20}=30\,\%$ ; $\dfrac{5}{20}=25\,\%$ ; $\dfrac{3}{20}=15\,\%$ . Total: $100\,\%$ .

Step 2: Sum of the products: $2\times0+4\times1+6\times2+5\times3+3\times4=0+4+12+15+12=43$ .

Step 3: Weighted mean: $\bar{x}=\dfrac{43}{20}=2.15$ .

Answer: on average, $2.15$ children per family.

Worked example

Problem: With the same data, compute the angles of the pie chart.

Step 1: One individual corresponds to $\dfrac{360^\circ}{20}=18^\circ$ .

Step 2: Angles: $2\times18^\circ=36^\circ$ ; $4\times18^\circ=72^\circ$ ; $6\times18^\circ=108^\circ$ ; $5\times18^\circ=90^\circ$ ; $3\times18^\circ=54^\circ$ .

Step 3: Check: $36+72+108+90+54=360$ .

Answer: $36^\circ$ , $72^\circ$ , $108^\circ$ , $90^\circ$ , $54^\circ$ .

Worked example

Problem: On the report card, Matondo has $14$ in mathematics (coefficient $4$ ), $12$ in French (coefficient $3$ ), $15$ in science (coefficient $2$ ) and $16$ in drawing (coefficient $1$ ). Compute the weighted mean.

Step 1: Products: $14\times4=56$ ; $12\times3=36$ ; $15\times2=30$ ; $16\times1=16$ .

Step 2: Sum of the products: $56+36+30+16=138$ . Sum of the coefficients: $4+3+2+1=10$ .

Step 3: $\bar{x}=\dfrac{138}{10}=13.8$ .

Answer: the mean is $13.8$ (the simple mean $\dfrac{57}{4}=14.25$ ignores the coefficients).

Watch out

Watch out: in a weighted mean, divide by the sum of the coefficients (or of the counts), not by the number of subjects or values. For Matondo, dividing $138$ by $4$ would give $34.5$ , an impossible mark out of $20$ . Always check that your mean lies between the smallest and the largest value.

Key points

Population, individual, variable (qualitative or quantitative), count, relative frequency $=\dfrac{\text{count}}{N}$ .
Stick chart or bar chart: height $=$ count.
Pie chart: angle $=\dfrac{\text{count}}{N}\times360^\circ$ .
Simple mean: sum $\div$ number of values.
Weighted mean: $\dfrac{\sum n_ix_i}{\sum n_i}$ .

✍️ Exercises

Answer one exercise at a time. Your answer is checked right away, with an explanation.

Exercise 1 of 8

★ Easy

Compute the mean of the numbers $12$ , $15$ , $9$ , $14$ , $10$ .

🎯 In this lesson, you will learn to

The lesson

👋 Introduction

📘 Explanation

📘 Explanation

📘 Explanation

✏️ Worked example

✏️ Worked example

✏️ Worked example

⚠️ Watch out

⭐ Key points

✍️ Exercises

Introduction

Explanation

Explanation

Explanation

Worked example

Worked example

Worked example

Watch out

Key points