Formatting numbers

Notation and tables

Roland Krause

MADS6

Tuesday, 10 December 2024

Material

Formating numbers

Three types of numbers

Exact numbers
Measured numbers

Numbers with an associated error

Computed numbers

Numbers resulting from (potentially complex) calculation of a measured quantity or simulation.

Note

In data science, we are frequently facing values with limited detail on their measurements. That’s not an invitation to ignore the fact.

Exact numbers

Counted objects

3 bikes
7 people
1015 items sold

Note

In text, we would write “three bikes and seven people”.

Exact relation

1km = 1000m

Speed of light

299 792 458 m/s (a definition)

Measured numbers

Data with natural errors

Multiple measurements will return differing results
Measurements by different people will return different results.
Precision is naturally limited

Why we care

In data science we typically do not measure data ourselves.

The value we print to a report however conveys a meaning of certainty.

Significant digits

Types of zeros

Leading zeros
| ||||||
0.000000230453400
          |    ||
Captive zero   Trailing zeros

Significant zeros

All non-zero numbers are significant.
Zeros surrounded by other numbers are significant.
Leading zeros are never significant
Trailing zeros after a decimal if they have been measured.

Examples

Value	Digits	Explanation
1642 m	4	All non-zero
10.303 ml	5	Captive zeros
67.0 g	3	Trailing measured zero
0.00053503	5	All leading zeros irrelevant

Your turn

How many significant digits?

1200001
12.000
0.0000000001
0.0010000001

Solution

Value	Digits	Explanation
1200001 m	7	Mostly captive zeros
12.000 g	5	If measured
0.0000000001	1	All leading zeros irrelevant
0.0010000001	8

How to display numbers

Many numbers in data science are not in our “nice” range from 0 - 1000.

Example numbers

my_val <- (c(0.1, 0.00000000123, 13/150660001, 761231, -3.243) )
my_val

[1]  1.00000e-01  1.23000e-09  8.62870e-08  7.61231e+05
[5] -3.24300e+00

Numbers printed to the console can be deceiving

print(1)

[1] 1

print(1.00001)

[1] 1.00001

print(c(1, 1.00001))

[1] 1.00000 1.00001

But you can’t say Base R is doing a bad job…

Common notations of numbers

Decimal notation

sprintf("%f", my_val)

[1] "0.100000"      "0.000000"      "0.000000"     
[4] "761231.000000" "-3.243000"

Scientific notation

Should be \(1.9 \times 10^{-4}\) but we usually get to see 1.9e-4.

$1.9 \times 10^{-4}$

sprintf("%e", my_val)

[1] "1.000000e-01"  "1.230000e-09"  "8.628700e-08" 
[4] "7.612310e+05"  "-3.243000e+00"

Engineering notation

Exponents are a multiple of three.

num(my_val, notation = "eng")

<pillar_num(eng)[5]>
[1] 100   e-3   1.23e-9  86.3 e-9 761.  e+3  -3.24e+0

Using the pillar::num()

Formating the C-style `sprintf()`

Formating

%i - Integer values
%f - Decimal numerical format (fixed)
%e - Scientific notation %E with capital E
%g - Best of both worlds: decimal if exponent is < -4.

Many other options for padding, currency symbols

Examples

sprintf("%f", my_val)

[1] "0.100000"      "0.000000"      "0.000000"     
[4] "761231.000000" "-3.243000"

sprintf("%e", my_val)

[1] "1.000000e-01"  "1.230000e-09"  "8.628700e-08" 
[4] "7.612310e+05"  "-3.243000e+00"

sprintf("%G", my_val)

[1] "0.1"        "1.23E-09"   "8.6287E-08" "761231"    
[5] "-3.243"

sprintf("%i", my_val)

Error in sprintf("%i", my_val): invalid format '%i'; use format %f, %e, %g or %a for numeric objects

Python

Modern Python is using sprintf-style syntax in its str.format() function. Using the sprintf functions is deprecated.

Check the documentation.

Rounding to significant digits

Display of numbers carries a meaning

Compare different numbers - Eight, 12.0, 1.76-e7, four, 3.0

Generally: Round to three significant digits

signif(my_val, 3)

[1]  1.00e-01  1.23e-09  8.63e-08  7.61e+05 -3.24e+00

Rounding measured data

Adding numbers need to be rounded to the least precise digit

[1] 11.961

[1] 12

Least precise value (4.5) carries a single digit.

Tables

Data in tables

Basic conventions

Text is left-aligned.
Numbers are aligned at the decimal place

Base R is doing this nicely but the result is not publication quality.

# A tibble: 344 × 4
   species island    bill_length_mm bill_depth_mm
   <fct>   <fct>              <dbl>         <dbl>
 1 Adelie  Dream               32.1          15.5
 2 Adelie  Dream               33.1          16.1
 3 Adelie  Torgersen           33.5          19  
 4 Adelie  Dream               34            17.1
 5 Adelie  Torgersen           34.1          18.1
 6 Adelie  Torgersen           34.4          18.4
 7 Adelie  Biscoe              34.5          18.1
 8 Adelie  Torgersen           34.6          21.1
 9 Adelie  Torgersen           34.6          17.2
10 Adelie  Biscoe              35            17.9
# ℹ 334 more rows

Playing with `gt`

penguins |>
  select(species, island, contains("bill")) |>
  sample_n(10) |>
  gt()

species	island	bill_length_mm	bill_depth_mm
Gentoo	Biscoe	44.5	14.3
Adelie	Torgersen	38.6	21.2
Gentoo	Biscoe	45.3	13.7
Chinstrap	Dream	52.8	20.0
Adelie	Torgersen	37.3	20.5
Chinstrap	Dream	43.2	16.6
Gentoo	Biscoe	47.5	14.2
Gentoo	Biscoe	52.2	17.1
Chinstrap	Dream	50.8	19.0
Gentoo	Biscoe	46.1	13.2

Playing with `gt`

penguins |>
  select(species, island, contains("bill")) |>
  sample_n(10) |>
  gt() |>
  cols_align(
    align = "left", 
    columns = c(species, island)) |>
  cols_label(
    species = "Species",
    island = "Island",
    bill_length_mm =  "Bill length (mm)"  ,
    bill_depth_mm = "Bill depth (mm)"
  )

Species	Island	Bill length (mm)	Bill depth (mm)
Adelie	Dream	35.6	17.5
Gentoo	Biscoe	55.9	17.0
Gentoo	Biscoe	43.2	14.5
Adelie	Torgersen	37.2	19.4
Adelie	Torgersen	34.6	17.2
Adelie	Dream	36.0	18.5
Chinstrap	Dream	52.2	18.8
Adelie	Dream	32.1	15.5
Adelie	Dream	37.2	18.1
Adelie	Biscoe	38.1	17.0

Playing with `gt`

penguins |>
  select(species, island, contains("bill")) |>
  sample_n(10) |>
  gt() |>
  cols_align(
    align = "left", 
    columns = c(species, island)) |>
  cols_label(
    species = "Species",
    island = "Island",
    bill_length_mm =  "Length"  ,
    bill_depth_mm = "Depth"
  ) |>
  tab_spanner("Bill dimensions (mm)", contains("bill"))

Species	Island	Bill dimensions (mm)
Species	Island	Length	Depth
Chinstrap	Dream	52.0	18.1
Adelie	Torgersen	35.9	16.6
Adelie	Torgersen	39.7	18.4
Adelie	Torgersen	42.8	18.5
Adelie	Biscoe	37.6	17.0
Adelie	Dream	40.2	17.1
Gentoo	Biscoe	45.5	15.0
Gentoo	Biscoe	46.1	13.2
Chinstrap	Dream	58.0	17.8
Chinstrap	Dream	46.5	17.9

Playing with `gt`

penguins |>
  select(species, island, contains("bill")) |>
  sample_n(10) |>
  gt() |>
  cols_align(
    align = "left", 
    columns = c(species, island)) |>
  cols_label(
    species = "Species",
    island = "Island",
    bill_length_mm =  "Length"  ,
    bill_depth_mm = "Depth"
  ) |>
  tab_spanner("Bill dimensions (mm)", contains("bill")) |> 
  tab_options(column_labels.background.color = "#00A4E1")

Species	Island	Bill dimensions (mm)
Species	Island	Length	Depth
Chinstrap	Dream	49.6	18.2
Adelie	Dream	37.6	19.3
Gentoo	Biscoe	45.4	14.6
Gentoo	Biscoe	51.3	14.2
Gentoo	Biscoe	43.8	13.9
Chinstrap	Dream	52.8	20.0
Adelie	Dream	40.9	18.9
Adelie	Torgersen	NA	NA
Adelie	Torgersen	41.1	17.6
Adelie	Torgersen	35.9	16.6

Playing with `gt`

penguins |>
  select(species, island, contains("bill")) |>
  sample_n(10) |>
  gt() |>
  cols_align(align = "left", columns = c(species, island)) |>
  cols_label(
    species = "Species",
    island = "Island",
    bill_length_mm =  "Length"  ,
    bill_depth_mm = "Depth"
  ) |>
  tab_spanner("Bill dimensions (mm)", contains("bill")) |> 
  tab_options(column_labels.background.color = "#00A4E1") |> 
  tab_style(
    style = cell_text(size = pct(120)),
    locations = cells_body()) |> 
  tab_style(
    style = cell_text(weight = "bold"),
    locations = list(cells_column_labels(),
                     cells_column_spanners()))

Species	Island	Bill dimensions (mm)
Species	Island	Length	Depth
Chinstrap	Dream	51.3	19.9
Gentoo	Biscoe	45.4	14.6
Adelie	Torgersen	36.2	17.2
Adelie	Biscoe	43.2	19.0
Gentoo	Biscoe	46.5	13.5
Adelie	Biscoe	41.1	18.2
Gentoo	Biscoe	45.5	13.9
Adelie	Biscoe	35.5	16.2
Adelie	Dream	36.0	17.1
Chinstrap	Dream	50.2	18.8

Before we stop

We looked into …

Some basic principles
Reviewed on material
Clean up a chart
Formatted numbers

Formatting numbers

Material

Formating numbers

Three types of numbers

Exact numbers

Counted objects

Exact relation

Speed of light

Measured numbers

Data with natural errors

Significant digits

Types of zeros

Significant zeros

Examples

Your turn

How many significant digits?

Solution

How to display numbers

Example numbers

Numbers printed to the console can be deceiving

Common notations of numbers

Decimal notation

Scientific notation

Engineering notation

Formating the C-style sprintf()

Formating

Examples

Rounding to significant digits

Display of numbers carries a meaning

Generally: Round to three significant digits

Rounding measured data

Tables

Data in tables

Basic conventions

Playing with gt

Playing with gt

Playing with gt

Playing with gt

Playing with gt

Before we stop

We looked into …

Acknowledgements

Formating the C-style `sprintf()`

Playing with `gt`

Playing with `gt`

Playing with `gt`

Playing with `gt`

Playing with `gt`