# An Adventure in Statistics: The Reality Enigma

## Student Resources

# Jig:Saw’s puzzle solutions

## Puzzle 1

Describe the following terms: frequency, relative frequency, proportion and percentage.

- Frequency is the number of times that a score, range of scores or category occurs.

- Relative frequency is the frequency of a score, range of scores or category expressed relative to the total number of observations:

- Relative frequencies are proportions. A proportion in statistics usually quantifies the portion of all measured data in a particular category in a
*scale of measurement*. In other words, it is the frequency of a particular score/category relative to the total number of scores:

- We can convert a relative frequency or proportion to a percentage by multiplying it by 100:

## Puzzle 2

Draw the frequency distribution of the RAS scores (with scores not grouped by class intervals).

Your hand drawn graph should look something like the one in Figure 1.

**Figure 1: Frequency distribution of RAS scores**

## Puzzle 3

In this chapter Zach looked at 20 women’s ratings of how important certain characteristics are in romantic partners. Here are the data for the characteristic ‘wants to have children’: 1, 1, 9, 1, 10, 3, 7, 6, 7, 2, 2, 9, 8, 2, 8, 6, 9, 2, 9, 6. Produce a frequency table of these data that includes:

The answers to this puzzle can be found in Table 1.

**Table 1: Frequency distribution of females’ ratings of ‘wants to have children’ as a characteristic in a romantic partner, including frequencies, relative frequencies, percentages, cumulative percentages and cumulative percentages**

## Puzzle 4

For the data in the previous question, remembering that scores of 0–5 mean ‘unimportant’ and 6–10 mean ‘important’, what percentage of adolescent women thought that it was important that their partners wanted to have children in the future?

If we look at the ‘cumulative percentage (down)’ column in the table above, we can see that the cumulative percentage of the response of 6 is 60%. Therefore, 60% of adolescent women thought that it was important that their partners wanted to have children in the future.

## Puzzle 5

Zach was worried that he was unappealing to women because he dropped out of college. Here are the ratings of the 20 women in the chapter for the characteristic ‘finished education’: 9, 8, 5, 4, 7, 3, 10, 7, 6, 4, 4, 8, 9, 1, 7, 3, 7, 6, 10, 9. Draw a histogram of these data. Do you think most women think that it is important that their relationship partner finished their college education?

Your hand-drawn histogram should look like Figure 2.

**Figure 2: Histogram of 20 women’s ratings of the importance of a relationship partner having finished their education**

Looking at the histogram, we can see that the scores are spread out across the whole scale, there is no accumulation of scores at either end of the scale. This indicates that the women did not consistently rate the characteristic ‘finished education’ as either important or unimportant – their views were quite varied.

## Puzzle 6

The polygon in Figure 3.10 (in the book and reproduced below) shows the ratings for the characteristic ‘romantic’. From this image reconstruct the raw data.

**Figure 3.10 (reproduced): Polygon of the ratings for the characteristic ‘romantic’**

Looking at the polygon, we can see that no one gave a rating of 0, 1, 2 or 6. One woman gave a rating of 3, one gave a 4 and one a 5. Eight women gave a rating of 7, four women gave a rating of 8, two women gave a rating of 9 and three women gave a rating of 10. Therefore, the raw data for the 20 women are: 3, 4, 5, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10.

## Puzzle 7

Here are the ratings for the same 20 women for the characteristic ‘attractive appearance’: 4, 10, 9, 8, 7, 8, 10, 8, 7, 3, 9, 10, 8, 10, 7, 9, 9, 9, 8, 7. Draw a frequency distribution of these scores.

Your hand-drawn frequency distribution should look like Figure 3

**Figure 3: Histogram of 20 women’s ratings of the importance of an attractive appearance in a relationship partner**

## Puzzle 8

Here are the ratings for the same 20 women for the characteristic ‘creativity’: 7, 6, 5, 4, 5, 8, 9, 5, 5, 7, 4, 5, 5, 10, 7, 3, 5, 9, 1, 7. Draw a frequency polygon of these scores.

Your hand-drawn frequency polygon should look like Figure 4.

**Figure 4: Frequency polygon of 20 women’s ratings of the importance of creativity in a relationship partner**

## Puzzle 9

The histogram in Figure 3.11 (in the book and reproduced below) shows the ratings for the characteristic ‘honesty’. From this image reconstruct the raw data.

**Figure 3.11 (reproduced): Histogram of the ratings for the characteristic ‘honesty’**

Looking at the histogram, we can see that no women gave ratings of 1, 2, 3, 4, 5, 6 or 7 because there are no bars above 0 for any of these ratings. Four women gave a rating of 8, four gave a rating of 9 and twelve gave a rating of 10. Therefore, the raw data for the 20 women are: 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10.

## Puzzle 10

Based on the histograms and polygons for the previous three puzzles, what characteristic do women most consistently find important in a romantic partner: attractive appearance, creativity or honesty? Explain your answer.

Looking back at the three graphs we can see that honesty is the characteristic that most women find consistently important in a romantic partner. We know this because the bars on the graph are clustered at the high end of the scale, all women gave honesty a rating of between 8 and 10, and over half of women gave it a rating of 10. Attractive appearance also received mostly high ratings of between 7 and 10 but the ratings are more evenly spread over this range than for the honesty characteristic. The ratings for creativity, on the other hand, were more spread out across the whole scale. The highest number of ratings occurred around the middle of the scale, suggesting that women did not consistently agree on the importance of this characteristic in a romantic partner, and most women rated it as neither important nor unimportant.

## Puzzle 11

Sketch the shape of a normal distribution.

Your graph should look like the one below (Figure 3.7 in the book).

**Figure 3.7 (reproduced): A ‘normal’ distribution (the curve shows the idealized shape)**

## Puzzle 12

Look at the histograms in Figure 3.1 (in the book and reproduced below). For each one comment on:

**Figure 3.1 (reproduced): Histograms of the importance to teenage women of four characteristics in their partners**

**a. How symmetrical you think they are**

The graphs for ‘high salary’ and ‘ambitious’ both look fairly symmetrical. The graph for ‘humour’ is less symmetrical and the graph for ‘kind’ is not symmetrical at all.

**b. How flat or pointy they are**

The histogram for ‘high salary’ is very flat: the bars are of quite similar heights across all of the categories. In fact, each point along the scale was endorsed by a similar number of women, which suggests that there is no consistent pattern in women valuing a high salary as an important characteristic in this sample.

The histogram for ‘kind’ is very pointy at the high end of the distribution. This suggests that most women rated kindness as an important characteristic in a romantic partner.

The histograms for ‘humour’ and ‘ambitious’ are not particularly pointy or flat, although the one for ‘humour’ certainly has a heavy tail at the top end of the scale (i.e., more scores than you might expect). It is interesting that the vast majority of women put their response above 5, which means they thought these characteristics were important to some extent.

**c. How skewed they are, and whether the skew is positive or negative**

The histogram for ‘high salary’ isn’t very skewed at all; the scores are evenly spread out across the whole scale, giving a very flat distribution. The histogram for ‘kind’, on the other hand, is highly negatively skewed; we can tell this because the frequent scores are clustered at the higher end and the tail points towards the lower, or more negative, scores. The histogram for ‘humour’ shows a slight negative skew. The histogram for ‘ambitious’ isn’t particularly skewed but, if anything, it shows slight positive skew.

These patterns indicate that kindness was consistently rated as a highly important characteristic in a romantic partner, as was humour. Ambitious was also consistently rated as an important characteristic, but to a lesser extent than kindness and humour. However, there was no consistent pattern for the characteristic ‘high salary’: about as many women rated this characteristic as highly important as rated it as unimportant.