Research methods in Cog Sci - 1-2

What are the 3 measures of central tendency?

• Whats different about nominal measurements?

Mean - avg - (Sum of values ÷ n)

Median - Middle of the ordered values (halfway between the middle two

if N is even)

Mode - most frequent value

• Only mode applies to nominal measurements 

  • No numbers - names 

  • The most frequent category in data set 

What are the measures of variability

Range

Variance

Standard

X

value 

• One point in data set

N

sample size 

• Whatever number of ppl you are testing on 

Ʃ

sigma 

• sum of numbers

μ

 mean 

• average of all points of data

X-μ

deviation from mean (value - mean)

• Distance of each individual score from the average/mean 

(X-μ)2

squared deviation (x minus mean, squared)

Ʃ (x-μ):

Sum of absolute deviation 

• Always equals 0 

• Balance out the data 

• total distance of all points above the mean perfectly cancels out the total distance of all points below the mean.

Ʃ (x-μ)2

sum of squared deviations from mean 

• measure how spread out a set of numbers is around their average value.

• measures the total amount of variability or spread in a set of data

Ʃ (x-μ)/N

Mean of absolute deviation 

• measures the average distance between each data point and the mean of a data set, showing how spread out or varied your numbers are

Ʃ (x-μ)2 /N

 (variance)  Mean of squared deviation 

• Variance measures how spread out a set of numbers is from their average value.

  • whether your data points are closely clustered around the center or scattered far apart

• √ Ʃ (x-μ)2 /N

Population SD = √variance

√ Ʃ (x-μ)2 /n-1

• Uses n-1 to compute mean instead of n 

• Sample SD= 𝝈 =√ Ʃ (x-μ)2 /n-1

x-μ/ 𝝈

Z score

• Value - mean/ standard deviation 

• z-score describes a single observation. 

• not a summary of a whole distribution, so it isn't a measure of variability

Discuss: Same data, two spread values. What should a reader ask?

• 73 points in course A sits 2 standard deviations above the mean.

• 73 points in course A sits 0.2 standard deviations above the mean

Consider - to properly evaluate a value you should consider how extreme it is compared to the population it belongs to 

Covariance

the direction of the linear relationship between two variables

(whether they move together or in opposite directions).

• cov(X, Y) = sum of [(x − mean of X) × (y − mean of Y)] ÷ N

Pearson’s r

describes both the direction and the exact strength of that relationship on a standardized scale from -1 to +1

• r = cov(X, Y) ÷ (SD of X × SD of Y)

Pearsons r

This normalizes the covariance so that r always falls between -1 and + 1 

+1: a perfect positive linear relationship −1: a perfect negative linear relationship 0: no linear relationship

Frequency

Frequency (count): how many times a value, or values within a range, appear.

• Ex: 48 of 100 people commuted by car.

Relative frequency: the count as a proportion or percentage of the total. 

• The same example gives 48%.

Cumulative frequency: the running total of counts up to a given value. For example, how many people scored 4 or below.

These are usually organized in a frequency table and displayed as a bar chart for

categorical data or a histogram for numerical data

histogram

 counts how many values fall in each bin, revealing the shape and range

2 variable scatter plot

• Each dot - one participant with 2 scores 

• Positive - as X goes up, Y goes up - same direction 

• Negative - as X goes up, Y goes down - diff directions 

• None - changes in X unrelated to changes in Y