The effect of changing the data

Statistics and probability, grade 10 (4 units) Lesson 12 of 20Learning

The effect of changing the data

Adding a number to all the data

When the same number is added to all the data, every measure of centre grows by that number, and the measures of spread do not change: all the data move together, and the distances between them stay.

The hourly pay of six temporary workers, in shekels, in order from least to greatest:
45, 48, 50, 52, 52, 53.
In April each of them got a rise of 4 ₪:
49, 52, 54, 56, 56, 57.

	Before	After
Mean	50	54
Median	51	55
Mode	52	56
Range	8	8
Standard deviation	$\displaystyle \approx2.77$	$\displaystyle \approx2.77$

The deviations from the mean did not change – $\displaystyle -5, -2, 0, 2, 2, 3$ – so neither did the variance and the standard deviation. The same holds when a number is taken away from all the data.

Multiplying all the data by a number

When all the data are multiplied by the same positive number $\displaystyle k$ , every measure of centre, and the range and the standard deviation too, are multiplied by $\displaystyle k$ . The variance is multiplied by $\displaystyle k^2$ .

Suppose that instead of a rise of 4 ₪, the pay of each of the six workers went up by 10%: each wage is multiplied by $\displaystyle 1.1$ .

The mean: $\displaystyle {50\cdot1.1}={55}$ ₪; the median: $\displaystyle {51\cdot1.1}={56.1}$ ₪; the mode: $\displaystyle {52\cdot1.1}={57.2}$ ₪.
The range: $\displaystyle {8\cdot1.1}={8.8}$ ₪.
The standard deviation: $\displaystyle 2.77\cdot1.1\approx3.05$ ₪. Every deviation from the mean grew 1.1 times, so the standard deviation did too.

Dividing all the data by a number is multiplying by $\displaystyle \frac{1}{k}$ : all these measures become $\displaystyle k$ times smaller.

The effect of the changes in a table

A change of all the data	Mean, median, mode	Range, standard deviation
Adding $\displaystyle c$	grow by $\displaystyle c$	do not change
Taking away $\displaystyle c$	shrink by $\displaystyle c$	do not change
Multiplying by $\displaystyle k>0$	are multiplied by $\displaystyle k$	are multiplied by $\displaystyle k$
Dividing by $\displaystyle k>0$	are divided by $\displaystyle k$	are divided by $\displaystyle k$

A change in two steps goes through both steps: degrees Celsius become degrees Fahrenheit by $\displaystyle {F}={1.8C+32}$ , so a mean of $\displaystyle 20^\circ$ Celsius is $\displaystyle {1.8\cdot20+32}={68^\circ}$ Fahrenheit, and a standard deviation of $\displaystyle 5^\circ$ Celsius is $\displaystyle {1.8\cdot5}={9^\circ}$ Fahrenheit – the 32 does not change it.

Adding data

When a new datum is added, the mean moves towards it, and the standard deviation depends on its distance from the mean.

The data 6, 8, 10, 12, 14: the mean is 10, the variance 8, and the standard deviation $\displaystyle \sqrt{8}\approx2.83$ .

Add 10, equal to the mean: the mean stays 10. The squared deviations still add up to 40, but they are divided by 6, so $\displaystyle {\sigma}={\sqrt{\frac{40}{6}}\approx2.58}$ – the standard deviation shrinks.
Add 16, far from the mean: the mean goes up to $\displaystyle {\frac{66}{6}}={11}$ , and $\displaystyle {\sigma}={\sqrt{\frac{70}{6}}\approx3.42}$ – the standard deviation grows.

A new datum equal to the median does not change the median when the number of data is odd; a new datum greater than the greatest value makes the range larger.

Building data from given measures

Sometimes the data themselves can be found from the measures.

Four boys work at the same place. Their pay for one working day:
The mean pay: 250 ₪.
The mode: 300 ₪.
The median: 265 ₪.

The solution below finds the four wages, in order from least to greatest: $\displaystyle a\le b\le c\le d$ .

The mode is 300: at least two of the wages are 300 ₪. The median, 265, is less than 300, so they are the two greatest: $\displaystyle {c}={d}={300}$ .
The median is the mean of the two middle ones: $\displaystyle {\frac{b+300}{2}}={265}$ , so $\displaystyle {b}={230}$ .
The mean is 250, so the wages add up to $\displaystyle {4\cdot250}={1000}$ : $\displaystyle {a}={1000-230-300-300}={170}$ .

The wages: 170, 230, 300, 300 ₪.

Exercise 1

Six students’ marks in a quiz, out of 20:
6, 7, 7, 8, 9, 11.
The mean is 8, the median 7.5, and the standard deviation about 1.63.
The teacher found that one of the questions was wrong, and added 2 points to every student.

What is the new mean?

/ – 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