🇨🇩 Data handling

Frequencies, charts and means

⏱ 30 minOfficial programme: CD-E8-STA-01

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🎯 In this lesson, you will learn to

Define population, variable, frequency, mode.
Draw bar, histogram, pie charts; simple and weighted means.

The lesson

Introduction

The student committee of a school in Bukavu wants to know how students travel to school, how long their journey takes and what marks they got in the last test. To answer, the data must be organised: counted, arranged in a table, shown in a chart and summarised by a mean. In this lesson you learn the vocabulary of statistics, you draw bar charts, histograms and pie charts, and you compute simple and weighted means.

Explanation

Vocabulary

Population: the set being studied (the students of the school); each element is an individual.
Variable: what is observed. It is qualitative (means of transport) or quantitative (mark, duration). A quantitative variable is discrete if it takes separate values (number of brothers and sisters), continuous if it can take any value in an interval (height); the values are then grouped into classes, such as $[20\,;30[$ .
Frequency (count) of a value: number of individuals with that value. Total $N$ : sum of the counts.
Relative frequency: $f=\dfrac{\text{count}}{N}$ , written as a fraction, a decimal or a percentage.
Mode: the value with the greatest count (for classes, we speak of the modal class).

Explanation

Charts

Bar chart: for a discrete variable. The values go on the horizontal axis; each value has a bar whose height is proportional to its count.
Histogram (in French diagramme en bandes): for classes. Each class is shown by a rectangle standing on its interval; the rectangles touch each other. When the classes have the same width, the height is proportional to the count.
Pie chart: to show the share of each category in the total. The angle of each sector is proportional to the count:
$\text{angle}=f\times360^\circ=\dfrac{\text{count}}{N}\times360^\circ$

Explanation

Means

Simple mean of $n$ values: $\bar{x}=\dfrac{x_1+x_2+\dots+x_n}{n}$ .
Weighted mean: when value $x_i$ has count (or weight) $n_i$ ,
$\bar{x}=\dfrac{n_1x_1+n_2x_2+\dots+n_px_p}{n_1+n_2+\dots+n_p}$
Grouped data: replace each class by its midpoint, for example $25$ for $[20\,;30[$ , then compute a weighted mean. The result is an approximate value of the mean.

Worked example

Problem: Here are the marks out of $10$ of $25$ students in a quiz.

Mark	$4$	$5$	$6$	$7$	$8$
Count	$2$	$5$	$8$	$6$	$4$

Find the relative frequency of the mark $6$ , the mode and the mean.

Step 1: $N=2+5+8+6+4=25$ . Relative frequency of $6$ : $\dfrac{8}{25}=0.32=32\%$ .

Step 2: The greatest count is $8$ : the mode is $6$ .

Step 3: $\bar{x}=\dfrac{2\times4+5\times5+8\times6+6\times7+4\times8}{25}=\dfrac{8+25+48+42+32}{25}=\dfrac{155}{25}=6.2$ .

Answer: relative frequency $32\%$ , mode $6$ , mean $6.2$ out of $10$ .

Worked example

Problem: Out of $40$ students, $22$ walk to school, $6$ cycle, $8$ come by motorbike taxi and $4$ by bus. Compute the angles of the pie chart.

Step 1: Walking: $\dfrac{22}{40}\times360^\circ=198^\circ$ .

Step 2: Bicycle: $\dfrac{6}{40}\times360^\circ=54^\circ$ ; motorbike taxi: $\dfrac{8}{40}\times360^\circ=72^\circ$ ; bus: $\dfrac{4}{40}\times360^\circ=36^\circ$ .

Step 3: Check: $198+54+72+36=360$ .

Answer: $198^\circ$ , $54^\circ$ , $72^\circ$ and $36^\circ$ .

Worked example

Problem: In the school garden, $20$ bean plants are measured (in cm). Compute the mean and give the modal class.

Class	$[10\,;20[$	$[20\,;30[$	$[30\,;40[$
Count	$4$	$9$	$7$

Step 1: Class midpoints: $15$ , $25$ and $35$ .

Step 2: $\bar{x}=\dfrac{4\times15+9\times25+7\times35}{20}=\dfrac{60+225+245}{20}=\dfrac{530}{20}=26.5$ cm.

Step 3: The greatest count is $9$ : the modal class is $[20\,;30[$ .

Answer: mean about $26.5$ cm; modal class $[20\,;30[$ .

Watch out

Common mistakes:

In a weighted mean, divide by the sum of the counts (or weights), not by the number of different values. In the marks example, dividing $155$ by $5$ would give $31$ , which is impossible for a mark out of $10$ .
The mode is a value of the variable, not a count: in the example, the mode is $6$ , not $8$ .

Key points

Population, individual, variable (qualitative or quantitative, discrete or continuous), count, relative frequency $=\dfrac{\text{count}}{N}$ .
Mode: value (or class) with the greatest count.
Bar chart for a discrete variable, histogram for classes, pie chart for shares: angle $=f\times360^\circ$ .
Weighted mean: $\dfrac{\sum n_ix_i}{\sum n_i}$ ; for classes, use the midpoints.

✍️ Exercises

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

Exercise 1 of 10

★ Easy

Which of these variables is quantitative?

🎯 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