Question

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question.

Peas in a Pod Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype codes, where 1 = smooth-yellow, 2 = smooth-green, 3 = wrinkled-yellow, and 4 = wrinkled-green. Can the measures of center be obtained for these values? Do the results make sense?

2 1 1 1 1 1 1 4 1 2 2 1 2 3 3 2 3 1 3 1 3 1 3 2 2

Short answer

(a) The mean is 1.9.

(b) The median is 2.0.

(c) The mode is 1.

(d) The midrange is 2.5.

The values for the measures of the center can be computed but interpreting them makes no sense.

Step-by-step solution

Given information

The phenotype codes of pea are listed as follows:

• smooth-yellow is coded as 1
• smooth-green is coded as 2
• wrinkled-yellow is coded as 3
• wrinkled-green is coded as 4

Compute mean

(a)

The formula for the mean is:

$\overline{x}=\frac{\sum x}{n}$where, xrepresents the values and n is the sample size.

Substitute the values in the formula.

role="math" localid="1643799208055" $$\begin{gathered} \overline{x}=\frac{2+1+...+2}{25} \\ =\frac{47}{25} \\ =1.88 \end{gathered}$$

Thus, the mean value is 1.9.

Compute median

(b)

The formula for the median is stated as follows:

• The middlemost observation is equal to the median if n is odd.
• The average of the two middlemost observations is equal to the median if n is even.

The sample size is 25.

Arrange the observations in ascending order.

1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
4

The middlemost observation is 2.

The median is given as:

$M=2$

Thus, the median is 2.0.

Compute mode

(c)

Mode is/are the observation(s) in the data with the highest repetitions.

The frequency distributions for the phenotypes are:

Phenotype	Frequency
1	11
2	7
3	6
4	1

The highest frequency is obtained for phenotype 1.

Thus, the mode of the data is 1.

Compute midrange

(d)

The formula for midrange is stated as:

$\text{Midrange}=\frac{\text{Minimum}\text{value}+\text{Maximum}\text{value}}{2}$

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{1+4}{2} \\ =\frac{5}{2} \\ =2.5 \end{gathered}$$

Thus, the midrange is 2.5.

State the meaningfulness of results

The data is recorded for four types of phenotypes. The phenotypes are categorical, which cannot be ordered. Thus, the data is nominal scaled data.

The measures of the center can be computed for the numerically expressed categories as shown above.

However, the measures for the center, except mode, are meaningless for nominal data and do not make any sense.