Comparing two sources of data

Data and probability, grade 8 Lesson 8 of 14Learning

Comparing two sources of data

Comparing by a measure of centre and by the range

To compare two groups of data, work out a measure of centre and the range of each, and compare them.

Two bus lines go from Shira’s home to school.
For six days Shira timed the ride on each line, in minutes.
Two series of data on one chart: two bars for each day, and a colour for each line.

Line 5, sorted: $\displaystyle 18, 18, 19, 20, 22, 35$
Line 12, sorted: $\displaystyle 21, 22, 23, 23, 24, 25$

	Mean	Median	Range
Line 5	22	19.5	17
Line 12	23	23	4

By the median, line 5 is faster: on most days its ride is shorter.
Line 5’s mean is higher than its median because of Wednesday, 35 minutes: an outlier.
Line 5’s range is much larger: there is no knowing in advance how long its ride will take.

The conclusion: line 5 is usually faster, and line 12 is more reliable. On a test day, when being late is not an option, line 12 is the one to take.

The same centre, a different spread

Two groups can have the same mean and the same median and still differ: the range shows the difference.

The marks of two groups of 9 pupils in a maths contest:

Group A: $\displaystyle 62, 70, 75, 78, 80, 82, 85, 88, 100$
Group B: $\displaystyle 74, 76, 78, 80, 80, 81, 82, 84, 85$

In both groups the mean is 80 ( $\displaystyle \frac{720}{9}$ ) and the median 80. But the range in group A is $\displaystyle {100-62}={38}$ , and in group B $\displaystyle {85-74}={11}$ : in group B all the pupils are at a similar level, while group A has very strong pupils and weaker ones.

A conclusion from data

A conclusion is only as good as the data it is built on. Before drawing one, check:

How many data are there? The prices of only two products are not enough to tell which shop is cheaper.
Who was asked? A survey on favourite sports held only at a football pitch does not describe all the pupils of the school.
When was it measured? One week’s temperatures in summer do not describe the weather all year.
How big is the difference? A mean of 81 against a mean of 80, in groups whose range is 30, is too small a difference to conclude anything from.

Exercise 1

The coach has to pick one player for the final.
The points of the two players in six games:

Ron: $\displaystyle 10, 25, 8, 30, 12, 11$
Adi: $\displaystyle 14, 16, 15, 17, 16, 18$

a

Work out each player’s mean.

Ron:

/ – 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

Adi:

/ – 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

b

Find each player’s median and range.

Ron: median

/ – 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

Ron: range

/ – 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

Adi: median

/ – 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

Adi: range

/ – 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

c

Whom would you pick? Explain using the measures you found.

Exercise 2

How many books the pupils of two classes read in a month, 25 pupils in each class:

a

Find the mode in each class.

8A

/ – 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

8B

/ – 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

b

Work out the mean number of books in each class.

8A:

/ – 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

8B:

/ – 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

c

In which class do the pupils read more? Explain.

Exercise 3

The highest temperature each day, in one week, at two weather stations:

a

Work out the mean temperature at each station.

Station A:

/ – 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

Station B:

/ – 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

b

Find the range at each station.
At which station does the weather change more from day to day? Explain.

Station A

/ – 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

Station B

/ – 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 4

Is the conclusion right? If the data are not enough to draw it, write what they lack.

Noga checked the prices of five products in two shops.
In shop A: 12, 8, 25, 6 and 9 shekels. In shop B: 10, 9, 22, 7 and 12 shekels.
The conclusion: shop B is cheaper, since its highest price is lower.
In a survey at a football club, 20 of 25 children answered that football is their favourite sport.
The conclusion: most children in the town like football more than any other sport.
On the 30 days of June, the mean highest temperature in one town was 31 degrees, and in another town 24 degrees.
The conclusion: in June the first town was hotter.

Formulas: data and probability, grade 8

Relative frequency

The relative frequency of an answer – $\displaystyle f$ its frequency, $\displaystyle n$ the total of all the frequencies
$\displaystyle \frac{f}{n}$

Measures of centre and the range

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The range – the largest value minus the smallest
$\displaystyle R=x_{\max}-x_{\min}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
An impossible event – 0, a certain event – 1
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$