1.3 Digits and Numbers
A digit is a symbol, while a number expresses an idea of quantity. Numbers are represented by digits, and it is essential to distinguish between these elements. See the Digit or number? article from The Britannica Dictionary.
Example 1.4 If there are 20 students in classroom A and 30 in classroom B, it can be said that there are \(20+30=50\) students in both classrooms. This is numerical information.
Example 1.5 If people in a group are labeled 0 for males and 1 for females, it is clear that 0 and 1 are not being treated as digits since they do not express quantities. Note, however, that this binary label is convenient in that its average equals the ratio of 1’s. If there are 5 women and 3 men, the proportion of women \(\pi_F\) in the group is \[\pi_F = \frac{1+1+1+1+1+0+0+0}{8} = \frac{5}{8} = 0.625\] The percentage is 62.5%.
The proportion equals the number of observations in a category divided by the total number of observations. It is a number between 0 and 1 that expresses the share of the observations in that category. The percentage is the proportion multiplied by 100. The sum of the proportions equals 1.00. The sum of the percentages equals 100. (Agresti and Franklin 2013, 25)
In this text, the American standard will be adopted, which uses the period symbol (.) as a decimal separator and a comma (,) as a thousands separator. Thus, \[\frac{1}{40} = 0.025 = 0.0250 = .025 = 2.5\% = \frac{2.5}{100}.\] Recurring decimals will be written in the form \(\frac{1}{3} = 0.333... = 0.\bar{3} \approx 0.333 \approx 0.3\). The number \(32,960 = 30,000 + 2,000 + 960\) should be read as ‘thirty-two thousand nine hundred and sixty’.
This option avoids many problems, since many statistical software are not compatible with the Brazilian standard, which uses a comma as a decimal separator and a point to separate thousands. In personal notes and lists of exercises, the notation preferred by the student may be adopted.
The numerical and monetary representations in the current location are detailed below. Check on your machine.
## Sys.localeconv()
## decimal_point .
## thousands_sep
## grouping
## int_curr_symbol USD
## currency_symbol $
## mon_decimal_point .
## mon_thousands_sep ,
## mon_grouping \003\003
## positive_sign
## negative_sign -
## int_frac_digits 2
## frac_digits 2
## p_cs_precedes 1
## p_sep_by_space 0
## n_cs_precedes 1
## n_sep_by_space 0
## p_sign_posn 1
## n_sign_posn 1