Population, sample and variable

Statistics and probability, grade 10 (4 units) Lesson 1 of 20Learning

Population, sample and variable

Population and sample

A population is all the items we want to learn about – people, houses, days, cars. A sample is the part of the population that is actually checked.

The student council wants to know how long the journey to school takes the school’s students.
The school has 600 students.
The council asked 120 of them.

The population: the school’s 600 students.
The sample: the 120 students who were asked.
A sample is checked when checking the whole population is impossible, or too expensive.
For a sample to teach something about the population, it has to be like it: 120 students from every year, not from one class only.

A variable and its values

A variable is a property checked in every item, which can differ from item to item. Everything the variable can take are the values of the variable.

The student council’s survey asked two questions:

“How many minutes does your journey to school take?” – the variable: the journey’s length; values such as 5 minutes, 12.5 minutes, 40 minutes.
“How do you get to school?” – the variable: the way of getting there; the values: on foot, by bike, by school bus, by car.

A datum is the value found for one item: one student’s journey takes 12 minutes, and he comes by bike.

The kinds of variables

The kind of variable decides what can be done with the data: work out a mean, put them in order, or only count.

A qualitative variable (also called nominal) – the values are names of groups, not numbers: eye colour, blood type, the way of getting to school. Their order and any arithmetic with them mean nothing.
A quantitative variable – the values are numbers that measure or count something:
- discrete – separate values, usually whole, that are counted: the number of people in a family, the number of brothers and sisters, the number of mistakes in a test.
- continuous – any value in a range is possible, and it is measured: height, weight, temperature, the journey’s length, the time of day.

A number does not make a variable quantitative: a phone number or a player’s shirt number is a name, so the variable is qualitative. Arithmetic with them means nothing.

The distribution of the data

The distribution of the data says which values came up and how many times each of them. The number of times a value came up is its frequency.

How the 120 students in the sample get to school:

on foot – 42 students;
by bike – 18;
by school bus – 36;
by car – 24.

The distribution shows the whole picture: the value that came up most often is “on foot”, and the frequencies together are the whole sample, $\displaystyle {42+18+36+24}={120}$ .

Exercise 1

For each variable write whether it is qualitative, quantitative discrete or quantitative continuous:

the number of messages a student sends in a day;
the time it takes to run 100 metres;
the favourite sport;
the shoe size;
the amount of rain that fell in a day, in millimetres;
the town you were born in.

Exercise 2

The council wants to know how many hours a week the town’s teenagers volunteer.
The town has 8,000 teenagers.
The council asked 400 of them.

What are the population, the sample and the variable? What kind of variable is it?

What percent of the population is the sample?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The number of children in 50 families in a neighbourhood was checked:

Number of children	0	1	2	3	4
Number of families	6	11	17	10	6

What is the variable, and what kind is it? What are the values of the variable?

How many children are there in all 50 families together?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Noa says: “An ID number is a number, so the variable ‘ID number’ is quantitative.”

Is Noa right? Explain.

Formulas: statistics and probability, grade 10 (4 units)

Open as a page

Presenting data

Relative frequency – $\displaystyle f$ a value’s frequency, $\displaystyle n$ the number of all the data
$\displaystyle \frac{f}{n}$
Relative frequency in percent
$\displaystyle \frac{f}{n}\cdot100\%$
The angle of a slice in a pie chart
$\displaystyle \frac{f}{n}\cdot360^{\circ}$

Measures of centre and position

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle x_1, \dots, x_k$ the values, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of data in classes – $\displaystyle m_1, \dots, m_k$ the classes’ middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The median’s place in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The lower decile’s place in an ordered list; the upper decile’s – $\displaystyle 0.9\cdot n$
$\displaystyle 0.1\cdot n$

Measures of spread

The range – the greatest value minus the least
$\displaystyle x_{max}-x_{min}$
The variance – the mean of the squared deviations from the mean
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}$
The standard deviation – the square root of the variance
$\displaystyle \sigma=\sqrt{\sigma^2}$
The variance from a frequency table
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2f_1+\dots+(x_k-\bar{x})^2f_k}{f_1+\dots+f_k}$

The rules of probability

Equally likely outcomes – $\displaystyle k$ the number of outcomes in the event $\displaystyle A$ , $\displaystyle n$ the number of outcomes in the sample space
$\displaystyle P(A)=\frac{k}{n}$
The probability of any event; $\displaystyle {P(\emptyset)}={0}$ , and $\displaystyle {P(U)}={1}$ for the sample space $\displaystyle U$
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
The union – $\displaystyle A$ or $\displaystyle B$ (or both); $\displaystyle A\cap B$ – the intersection, both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)-P(A\cap B)$
The union of disjoint events
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)$ $\displaystyle A\cap B=\emptyset$

Conditional probability and independence

The probability of $\displaystyle A$ given $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
A path in a tree – the product of the probabilities on its branches
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(B)\cdot P(A|B)$
The law of total probability – a weighted mean of the conditional probabilities
$\displaystyle P(A)=P(B)\cdot P(A|B)+P(\overline{B})\cdot P(A|\overline{B})$
Bayes’ formula – from $\displaystyle P(B|A)$ to $\displaystyle P(A|B)$
$\displaystyle P(A|B)\cdot P(B)$ $\displaystyle =$ $\displaystyle P(B|A)\cdot P(A)$
Independent events
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(A)\cdot P(B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)$ $\displaystyle =$ $\displaystyle P(A)$