Introduction

Common Designs

• Experimental
  • manipulates independent variable to see the effect it has on the dependent variable
• Correlational
  • does not manipulate anything
  • looks at the relationship between two variables

Populations & Samples

• A population is a large group of individuals, which a law of nature applies

• A sample is a group of a given population intended to represent that population

• Participants are those measured in a sample. Participants always refers to individuals (e.g., students, children, prisoners)

• A sample is supposed to generalize to a given population

• Participants > Subjects

Statistics vs Parameters

• Statistics use English letters to get values for a sample

• Parameter use Greek letters for values of a population

• Statistic = Sample

• Parameter = Population

Frequencies

• A raw score is the score given to a participant

• Frequency denoted as f

  • number of times a score occurs

• Below we see the amount of cars that have 4, 6, and 8 cylinders

table(mtcars$cyl)

## 
##  4  6  8 
## 11  7 14

• Note. Not F, that is a

• Frequency Distribution is a distribution of each score and the number of times the score has occurred

• We can use frequencies to get the proportion of cars that have 4, 6, and 8 cylinders

cyl_prop <- mtcars %>% 
  group_by(cyl) %>% 
  summarize(n = n(),
            prop = n/nrow(mtcars)) %>% 
  ungroup()

• We can see that there are mostly 8 cylinder cars (0.4375)

Visualizing Frequencies

• One reliable way of visualizing frequencies is by using a bar graph

• A Bar graph is a visualization with vertical bar over each nominal/ordinal category

mtcars %>% 
  ggplot(aes(as.factor(cyl))) + 
  geom_bar(fill = 'dodgerblue')

• Note A pie graph will always be inferior to a bar graph/any other visual

mtcars %>% 
  ggplot(aes(x = '',
             y = cyl,
             fill = as.factor(cyl))) + 
  geom_bar(stat = 'identity',
           width = 1) + 
  coord_polar('y', start = 0) +
  theme_void()

• A Histogram is a frequency visualization used primarily for interval or ratio scores

mtcars %>% 
  ggplot(aes(mpg)) + 
  geom_histogram(color = 'white',
                 fill = 'dodgerblue',
                 bins = 15)

• A Frequency Polygon is similar to a histogram, which shows data points connected with straight lines

mtcars %>% 
  group_by(cyl) %>% 
  summarize(n = n()) %>% 
  ungroup() %>% 
  ggplot(aes(cyl, n)) + 
  geom_line()

Normal Distribution

• The normal curve is often called the bell-shaped curve
  • Symmetrical
• A normal distribution is the same as the normal curve
  • Represents the population
• The distribution tails are the ends of the distribution
  • We’ll come back to these

Skewed Distributions

• Negative Skew is not normal and is asymmetrical

  • Indicates higher frequency of middle and higher scores

• Positive Skew is also not normal and is asymmetrical

  • Indicates higher frequency of low and middle scores

• Some thresholds for level of acceptable skewness is \(\pm2\) or \(\pm3\)

  • So if your value is smaller than this, you probably have acceptable skewness

• Kurtosis is when your frequency are really skinny and tall or really flat and wide

• A bimodal distribution is when your distribution has two medians (we’ll cover this shortly)

  • Has two distributions that should be addressed

Frequency Calculations

\[ Relative\;frequency = \frac{f}{N} \]

f <- 10
N <- 200

freq <- f/N
freq

## [1] 0.05

percent <- freq*100
percent

## [1] 5

Relative Frequency Using Normal Curve

• The proportion of area under the curve is the proportion of total area under the normal curve

• A percentile is the percentage of all scores in the sample below a particular score

• Cumulative Frequency is the number of scores in the data at or below a particular score

Central Tendency

• Three measures of central tendency

  • Mode
  • Median
  • Mean

• Measures of Central Tendency are statistics that summarize the location of a distribution on a variable by indicating where the center is

• A Normal distribution will have the central point right at the center

• A skewed distribution will have the central point where the frequency of scores is the highest

• The mode is the value with the highest frequency

calc <- c(3, 5, 4, 6, 6, 3, 5, 1, 8, 10)
calc

##  [1]  3  5  4  6  6  3  5  1  8 10

table(calc)

## calc
##  1  3  4  5  6  8 10 
##  1  2  1  2  2  1  1

• The value with the highest frequency is actually both 5 & 6

  • So for this example, we actually have two modes

• The median is the middle value

  • The 50th percentile

• Preferred for ordinal/ordered data

• More reliable for skewed data

• Calculating the median is taking the middle value in the order from lowest to highest scores

• We can see below that the middle value is between 6 and 3, but these are not sorted, so we can sort them to get the correct median

calc

##  [1]  3  5  4  6  6  3  5  1  8 10

• Sorted, we can see that the mode is between 5 and 5, which is 5
  • If other values are included, then you would get the average of the two middle values
  • In this case it is redundant but it would be \(\frac{5 + 5}{2} = 5\)

sort(calc)

##  [1]  1  3  3  4  5  5  6  6  8 10

• The Mean/Average

• The mean is the score located at the mathematical center of a distribution

  • Often denoted as \(Mean = \overline{X}\)

• The formula below is to calculate the mean

  • Where you would take all the values for each participant and divide by the number of participants

\[ \overline{X} = \frac{\Sigma\;X}{N} \]

• The mean is the basis for most inferential statistics

• I’ll use the values we randomly created for the mode

calc

##  [1]  3  5  4  6  6  3  5  1  8 10

sum_x <- sum(calc)
mean_n <- length(calc)

average <- sum_x/mean_n
average

## [1] 5.1

• We could also simply use the R base functions, like mean and/or median

median(calc)

## [1] 5

mean(calc)

## [1] 5.1

Deviation

• The distance a participant’s score/value is from the mean

• Deviations can be positive or negative

  • Participants can score lower (negative) than the mean and higher (positive) than the mean

• To get the deviation, you subtract the mean from each participant’s score

• X is the value that corresponds to each participant (their raw score) and \(\overline{X}\) is the average for the sample

\[ X - \overline{X} \]

• The larger the value the farther away from the mean the score/value is

• Sum of the deviations around the mean is the sum of all differences between the scores and the mean

In the Near Future

• Deviations is the start for upcoming lectures and statistical tests, especially the sum of the deviations

• \[ \Sigma(X - \overline{X}) \]

• If the sum of the deviations is 0 then that means your math is good

• Deviation of each score/value from the mean is often referred to as error/residual in statistical tests

set.seed(06062022)
devs <- tibble(x = rnorm(60, 5, n = 100)) %>% 
  rowid_to_column()

devs$mean <- mean(devs$x)
devs$deviations <- devs$x - devs$mean

head(devs)

## # A tibble: 6 x 4
##   rowid     x  mean deviations
##   <int> <dbl> <dbl>      <dbl>
## 1     1  73.2  60.0     13.2  
## 2     2  56.2  60.0     -3.78 
## 3     3  57.7  60.0     -2.27 
## 4     4  59.8  60.0     -0.172
## 5     5  60.9  60.0      0.909
## 6     6  61.8  60.0      1.76

• We can also visualize the deviations away from the average
  • We can see that some scores are farther from the average score, while some are extremely close to the mean

devs %>% 
  ggplot(aes(rowid, x)) + 
  geom_point(aes(color = deviations)) +
  geom_hline(yintercept = mean(devs$x),
             size = 1.25, 
             color = 'red')

• We can also get the sum of the deviations, by adding everything together
  • If our math is correct, it should be zero (or an extremely small value since R is a computer) to show that we calculated our deviations correctly

sum(devs$deviations)

## [1] -2.629008e-13