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

Sales of LP Vinyl Record Albums Listed below are annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of center give us any information about a changing trend over time?

0.3 0.6 0.8 1.1 1.1 1.4 1.4 1.5 1.2 1.3 1.4 1.2 0.9 0.9 1 1.9 2.5 2.8 3.9 4.6 6.1

Short answer

(a). The mean is 1.80 million.

(b). The median is 1.30 million.

(c). The mode is 1.4 million.

(d). The midrange is 3.20 million.

The data is in chronological order; thus, the values describe the change in trend by the non-symmetric distribution of values.

Step-by-step solution

Given information

The annual sales of vinyl record albums in the U.S. in millions of units are listed below.

The observations are recorded in chronological order.

0.3, 0.6, 0.8, 1.1, 1.1, 1.4, 1.4, 1.5, 1.2, 1.3, 1.4, 1.2, 0.9, 0.9, 1, 1.9, 2.5, 2.8, 3.9, 4.6, 6.1

Compute mean

(a)

For observation x, which are n in count, the mean is computed as:

$\overline{x}=\frac{\sum x}{n}$

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{0.3+0.6+0.8+...+6.1}{10} \\ =\frac{37.9}{21} \\ \approx 1.8048 \end{gathered}$$

Thus, the mean value is approximately 1.80 million.

Compute median

(b)

The median is computed as follows:

• When n is odd, the observation in the middle is the median.
• When n is even, the mean of the two middlemost observations is the median.

The number of observations is 21.

Arrange the observations in ascending order.

0.3
0.6
0.8
0.9
0.9
1
1.1
1.1
1.2
1.2
1.3
1.4
1.4
1.4
1.5
1.9
2.5
2.8
3.9
4.6
6.1

The middlemost observation is 1.3.

The median is given as:

$M=1.3$

Thus, the median is 1.30 million.

Compute mode

(c)

The mode is the value with the highest counts.

Observation	Frequency
0.3	1
0.6	1
0.8	1
0.9	2
1	1
1.1	2
1.2	2
1.3	1
1.4	3
1.5	1
1.9	1
2.5	1
2.8	1
3.9	1
4.6	1
6.1	1

Thus, the mode of the observations is 1.4 million.

Compute midrange

(d)

The midrange is the average of the minimum and maximum values.

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{0.3+6.1}{2} \\ =\frac{6.4}{2} \\ =3.2 \end{gathered}$$

Thus, the midrange is 3.20 million.

Explain if the measures of center can be used to describe the trend of the values.

Since the values are observed in chronological order, the measures of the center can be interpreted as per time periods.

For example, the mean describes that the overall average of all sales throughout the period is 1.80 million, which is lower than the midrange value of 3.2 million and larger than the median value of 1.3 million.

It implies there are more extreme values of sales in the most recent period compared to older ones.

Also, the median < mode< mean, which means the data is skewed (non-symmetric), inferring a discrepancy in the sale during the periods.