Question

Stocks and Sunspots. Listed below are annual high values of the Dow Jones Industrial Average (DJIA) and annual mean sunspot numbers for eight recent years. Use the data for Exercises 1–5. A sunspot number is a measure of sunspots or groups of sunspots on the surface of the sun. The DJIA is a commonly used index that is a weighted mean calculated from different stock values.

DJIA	14,198	13,338	10,606	11,625	12,929	13,589	16,577	18,054
Sunspot Number	7.5	2.9	3.1	16.5	55.7	57.6	64.7	79.3

1. Data Analysis Use only the sunspot numbers for the following.

a. Find the mean, median, range, standard deviation, and variance.

b. Are the sunspot numbers categorical data or quantitative data?

c. What is the level of measurement of the data? (nominal, ordinal, interval, ratio)

Short answer

a.

Mean: 35.91

Median: 36.10

Range: 76.40

Standard Deviation: 31.45

Variance: 989.10

b. The given sunspot numbers are quantitative data.

c.The level of measurement is the ratio scale.

Step-by-step solution

Given information

Data are given for two variables, “DJIA” and “Sunspot Number”.

Computation of statistics

a.

Let x be the sunspot number.

Mean:

The mean of the sample sunspot number is computed below:

\(\begin{aligned}{c}Mean &= \frac{{\sum x }}{n}\\ &= \frac{{7.5 + 2.9 + ..... + 79.3}}{8}\\ &= 35.91\end{aligned}\)

Therefore, the mean value is 35.91.

Median:

Arrange the sunspot numbers in increasing order:

2.9

3.1

7.5

16.5

55.7

57.6

64.7

79.3

The number of observations (n) is 8, which is an even number.

Thus, the median of the sample is computed as shown:

\(\begin{aligned} Median &= \frac{{{{\left( {\frac{n}{2}} \right)}^{th}}obs + {{\left( {\frac{n}{2} + 1} \right)}^{th}}obs}}{2}\\ &= \frac{{{{\left( {\frac{8}{2}} \right)}^{th}}obs + {{\left( {\frac{8}{2} + 1} \right)}^{th}}obs}}{2}\\ &= \frac{{{4^{th}}obs + {5^{th}}obs}}{2}\\ &= \frac{{16.5 + 55.7}}{2}\end{aligned}\)

\( = 36.10\)

Therefore, the median value is 36.10.

Range:

The range of the sunspot numbers is computed below:

\(\begin{array}{c}{\rm{Range}} = {\rm{Maximum}}\;{\rm{Value}} - {\rm{Minimum}}\;{\rm{Value}}\\ = 79.3 - 2.9\\ = 76.4\end{array}\)

Therefore, the range is 76.4.

Standard Deviation:

The sample standard deviation is computed below:

\(\begin{aligned} s &= \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \\ &= \sqrt {\frac{{{{(7.5 - 35.91)}^2} + {{(2.9 - 35.91)}^2} + ...... + {{(79.3 - 35.91)}^2}}}{{8 - 1}}} \\ &= 31.45\end{aligned}\)

Therefore, the standard deviation is 31.45.

Variance:

The sample variance is the square of the sample standard deviation.

Thus,

\(\begin{aligned} {s^2} &= {31.45^2}\\ &= 989.10\end{aligned}\)

Therefore, the variance is 989.10.

Categorical data vs. quantitative data

b.

Categorical Data: Data that consists of labels/tags and is not measurable is called qualitative/categorical data.

Quantitative Data: Data that can be counted/measured and consists of numbers is called numerical/quantitative data.

The sunspot numbers represent the number of spots on the surface of the sun. Since this is a measured numeric value, the data are quantitative.

Level of measurement

c.

There are four levels of measurement:

Nominal: Data that contains categories and cannot be arranged in any specific order is measured on a nominal scale.

Ordinal Data: Qualitative data that can be arranged in a specific order but the distances between successive values is not known is measured on an ordinal scale.

Interval: Numerical data that can be ordered and the difference between the values is known is measured on an interval scale. The natural zero point of the values is not well-defined, and the ratios of values cannot be computed.

Ratio: Numerical data whose ratios can be computed, and the natural zero point holds meaning is measured on a ratio scale.

The sunspot numbers are numeric. Their ratios can be computed, and the natural zero point is well-defined.

Here, 10 sunspot numbers are twice as large as fivesunspot numbers. Thus, ratios can be computed.

Zero represents no sunspot number and can be considered the starting point.

Thus, sunspot numbers are measured on a ratio scale.