New Zealand
Level 7 - NCEA Level 2

# Comparison of Scores

Lesson

When you get your marks back for an assessment, I bet you often ask your teacher for the average, or the mean mark. Let's take a look now at why it would also be beneficial for you to ask your teacher for the standard deviation of the results as well.

In the previous section we examined what a $z$z-score is and how to calculate them.

RECALL: Formula for Calculating z-scores from a Population

$z=\frac{x-\mu}{\sigma}$z=xμσ

This means:

$\text{standardised z score}=\frac{\text{raw score}-\text{population mean score}}{\text{standard deviation}}$standardised z score=raw scorepopulation mean scorestandard deviation

We also spoke a little about how you can use them to compare results. Let's delve into this a bit further.

## The tale of two maths tests

Let's say that you have completed two maths tests this year. In each test you scored $70%$70%. In each test the mean mark was $60%$60%.

At first glance it appears that you performed just as well in both tests when comparing yourself to the mean of the group. I'll now reveal the standard deviation of each test and now we'll get the true picture of your performance.

Let's say that the standard deviation for Test $1$1 was $10%$10% and the standard deviation for Test $2$2was $15%$15%.

We can see that for Test $1$1 you achieved a mark one whole standard deviation above the mean. Well done!

For Test $2$2 however, you achieved a mark less than one whole standard deviation above the mean, which indicates that your achievement in Test $2$2 is not as strong as your achievement in Test $1$1.

We can calculate the $z$z-scores to gain a little more insight.

$Z$Z-score for Test $1$1:

 $z$z $=$= $\frac{x-\mu}{\sigma}$x−μσ​ $=$= $\frac{70-60}{10}$70−6010​ $=$= $1$1

$Z$Z-score for Test $2$2:

 $z$z $=$= $\frac{x-\mu}{\sigma}$x−μσ​ $=$= $\frac{70-60}{15}$70−6015​ $=$= $0.66666$0.66666... $z$z $=$= $0.67$0.67 (to 2 d.p.)

Our calculations support our observations about the achievement in both maths tests.

#### Worked Examples

##### Question 1

Marge scored $43$43 in her Mathematics exam, in which the mean score was $49$49 and the standard deviation was $5$5. She also scored $92.2$92.2 in her Philosophy exam, in which the mean score was $98$98 and the standard deviation was $2$2.

1. Find Marge’s $z$z-score in Mathematics.

2. Find Marge’s $z$z score in Philosophy.

3. Which exam did Marge do better in, compared to the rest of her class?

Philosophy

A

Mathematics

B

Philosophy

A

Mathematics

B

##### question 2

A factory packages two types of cereal: Rainbow Crispies in a $600$600 g box and Honey Combs in a $650$650g. A box of Rainbow Crispies has a mean mass of $600$600 g with a standard deviation of $2.2$2.2 g. A box of Honey Combs has a mean mass of $650$650 g with a standard deviation of $1.4$1.4 g.

1. A box of Rainbow Crispies was selected at random for quality control. It had a mass of $613.2$613.2 g. Calculate the $z$z-score of this box.

2. A box of Honey Combs was selected at random for quality control. It had a mass of $653.08$653.08 g. Calculate the $z$z-score of this box.

3. Based on the z-scores, which box of cereal is closer to its marked mean mass.

Honey Combs

A

Rainbow Crispies

B

Honey Combs

A

Rainbow Crispies

B

### Outcomes

#### S7-4

S7-4 Investigate situations that involve elements of chance: A comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions B calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.

#### 91267

Apply probability methods in solving problems