01-descriptives-d.Rmd

# A full descriptive

To look at a full descriptive example including the mean, standard deviation, standard error, 95% Confidence Intervals, and z-scores, we will use the following data from 10 participants:

```{r we, message=FALSE, warning=FALSE, echo=FALSE}

dat_o <- tibble(Participants = 1:10,
              PreTest = c(60, 64, 56, 82, 74, 79, 63, 59, 72, 66),
              PostTest = c(68, 75, 62, 85, 73, 85, 64, 59, 73, 70))

dat <- dat_o %>% mutate(X = abs(PostTest - PreTest)) %>% select(Participants, X)

meanx <- dat %>% pull(X) %>% mean()
varx <- dat %>% pull(X) %>% var()
sdx <- dat %>% pull(X) %>% sd()
nx <- dim(dat)[1]
semx <- sdx/sqrt(nx)
zci <- 1.96

knitr::kable(dat, align = "c")
```

### The Mean

We know the formula for the mean is:

$$\overline{X} = \frac{\sum_i^n{x_i}}{n}$$

If we start to fill in the information from above we get:

$$\overline{X} = \frac{`r dat %>% filter(Participants == 1) %>% pull(X)` + `r dat %>% filter(Participants == 2) %>% pull(X)` + `r dat %>% filter(Participants == 3) %>% pull(X)` +`r dat %>% filter(Participants == 4) %>% pull(X)` +`r dat %>% filter(Participants == 5) %>% pull(X)` +`r dat %>% filter(Participants == 6) %>% pull(X)` +`r dat %>% filter(Participants == 7) %>% pull(X)` +`r dat %>% filter(Participants == 8) %>% pull(X)` +`r dat %>% filter(Participants == 9) %>% pull(X)` +`r dat %>% filter(Participants == 10) %>% pull(X)`}{`r dim(dat)[1]`}$$

And if we sum all the values on the top half together we get:

$$\overline{X} = \frac{`r dat %>% pull(X) %>% sum()`}{`r dim(dat)[1]`}$$

And finally divide the top half by the bottom half, leaving:

$$\overline{X} = `r (dat %>% pull(X) %>% sum())/(dim(dat)[1])`$$

We find that the mean, rounded to two decimal places, is **$\overline{X} = `r meanx`$**

### Standard Deviation

Next we want an interval estimate, most commonly the standard deviation. The formula for the standard deviation is:

$$s = \sqrt\frac{\sum_i^n(x_{i} - \overline{x})^2}{n-1}$$
So let's again start by filling in the numbers to the formula:

$$s = \sqrt\frac{(`r dat %>% filter(Participants == 1) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 2) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 3) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 4) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 5) %>% pull(X)` - `r meanx`)^2 + \\(`r dat %>% filter(Participants == 6) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 7) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 8) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 9) %>% pull(X)` - `r meanx`)^2 + (`r dat %>% filter(Participants == 10) %>% pull(X)` - `r meanx`)^2}{`r dim(dat)[1]` - 1}$$

And we do the subtractions in all those brackets of the top half and the one on the bottom:

$$s = \sqrt\frac{(`r dat %>% filter(Participants == 1) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 2) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 3) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 4) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 5) %>% pull(X) - meanx`)^2 + \\(`r dat %>% filter(Participants == 6) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 7) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 8) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 9) %>% pull(X) - meanx`)^2 + (`r dat %>% filter(Participants == 10) %>% pull(X) - meanx`)^2}{`r dim(dat)[1]-1`}$$

Then we square all those values on the top half:

$$s = \sqrt\frac{`r (dat %>% filter(Participants == 1) %>% pull(X) - meanx)^2` + `r (dat %>% filter(Participants == 2) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 3) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 4) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 5) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 6) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 7) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 8) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 9) %>% pull(X) - meanx)^2`+ `r (dat %>% filter(Participants == 10) %>% pull(X) - meanx)^2`}{`r dim(dat)[1]-1`}$$

Sum those values together:

$$s = \sqrt\frac{`r ((dat %>% filter(Participants == 1) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 2) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 3) %>% pull(X) - meanx)^2)  + ((dat %>% filter(Participants == 4) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 5) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 6) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 7) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 8) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 9) %>% pull(X) - meanx)^2) + ((dat %>% filter(Participants == 10) %>% pull(X) - meanx)^2)`}{`r dim(dat)[1]-1`}$$
Divide the top half by the bottom half:

$$s = \sqrt{`r varx`}$$

And finally take the square root:

$$s = `r sdx`$$

We find that the standard deviation, rounded to two decimal places, is **$s = `r sdx %>% round2(2)`$**

### Standard Error of the Mean

We also want to include a measure of how our sample relates to the population of interest and to do that we can use the Standard Error of the Mean and 95% Confidence Intervals. The formula for the standard error can be written as:

$$SE = \frac{SD}{\sqrt{n}}$$

We know that our $SD = `r sdx`$ and that we have $n = `r nx`$ participants, so:

$$SE = \frac{`r sdx`}{\sqrt{`r nx`}}$$

And if we do the square root of the bottom half (denominator):

$$SE = \frac{`r sdx`}{`r sqrt(nx)`}$$

And divide the top by the bottom:

$$SE = `r semx`$$

We find that the standard error, rounded to two decimal places, is $SE = `r semx %>% round2(2)`$

### Confidence Intervals

And we can use the standard error not to calculate the Upper and Lower 95% Confidence Intervals using the cut-off value (assuming $\alpha = .05$ and two-tailed) of $z = 1.96$. The key formulas are:

$$Upper \space 95\%CI = \overline{x} + (z \times SE)$$

And

$$Lower \space 95\%CI = \overline{X} - (z \times SE)$$

We know that:

* $\overline{x} = `r meanx`$
* $SE = `r semx`$
* $z = `r zci`$

**Upper 95% CI**

Dealing with the **Upper 95% CI** we get:

$$Upper \space 95\%CI = `r meanx` + (`r zci` \times `r semx`)$$

And if we sort out the bracket first:

$$Upper \space 95\%CI = `r meanx` + `r zci * semx`$$

Which leaves us with:

$$Upper \space 95\%CI = `r meanx + (zci * semx)`$$

**Lower 95% CI**

And now the **Lower 95% CI** we get:

$$Lower \space 95\%CI = `r meanx` - (`r zci` \times `r semx`)$$

And if we sort out the bracket first:

$$Lower \space 95\%CI = `r meanx` - `r zci * semx`$$

Which leaves us with:

$$Lower \space 95\%CI = `r meanx - (zci * semx)`$$

And if we round both those values to two decimal places, then you get **Lower 95%CI = `r (meanx - (zci * semx)) %>% round2(2)`** and **Upper 95%CI = `r (meanx + (zci * semx)) %>% round2(2)`**

### The Write-Up

And if we were to do a decent presentation of this data, including a measure of central tendency (the mean), a measure of spread relating to the sample (the standard deviation), and a measure of spread relating to the population (the 95% CI), then it would start as something like: **"We ran `r nx` participants (M = `r meanx %>% round2(2)`, SD = `r sdx %>% round2(2)`, 95%CI = [`r (meanx - (zci * semx)) %>% round2(2)`, `r (meanx + (zci * semx)) %>% round2(2)`])....."**