Lecture 2: Central Tendency and Variability

Interpolation

• estimating percentile ranks not defined

  • e.g. job wages estimations (eight hours work = $60, how many hours to get $30 = half or four hours)

How to solve percentile

• Solve each problem

• e.g. score, asked for percentile rank

• 30th percentile rank

  • You know that it is between 25 and 45 cumulative percentile (or the percentile percentage is the better def) on the graph

  • interval width = 10

  • percentile = 45 - 25 = 20

  • Intermediate value of the interval

    • e.g. 30 percentile rank, fraction = (45-30)/20 = 15/20 = 0.75 (or 75 percent of the way to 45) (between 25 to 45 ranking, it is 75 percent to 45)

  • Determine positions on the new scale

    • Distance = 0.75 ×10 = 7.5 units

  • Use distance from the top to determine new position: percentile = 29.5 - 7.5 = 22

---

• For 52 percentile

  • interval width = 10

  • percentile = 80-45 = 35

  • intermediate value = (80-52)/35 = 0.8

  • Position: 39.5 - 0.8 × 10 = 31.5

---

• What is the percentile rank for X = 46?

  • Interval width = 10

  • Percentile width = 20

  • Intermediate value: (49-46)/10 = 0.3

  • Position = 100-0.3×20 = 94 percentile

---

• What is the percentile rank for X = 21

  • In 20-29 interval = 10 width

  • percentile width =

Central Tendency

• tendency towards the center of the distribution

• An informative, representative of the population

  • Depends on the type of distribution you have

• Three major measures that have diff properties used in diff situation

  • Mean

  • Median

  • Mode

• In the normal distribution all three are very similar - expected, hopefully result

• Types of not normal = skew = three measures not similar and mean is not presentative of the results

  • Positive skew: mean is higher than the mode and median

    • Disproportionally more low values and less high values

  • Negative skew: mean is lower than the median and mode

    • Disproportionally more high end cases and fewer low cases

• Therefore, have to use the proper measure, rather than always going for the mean

Means

• main def

• population symbol and equation

---

Use of means

• the arithmetic average of all scores

  • Influenced by extreme scores and mean is easily affected

• Population: mu symbol

• Use for data in interval or ratio scales

• More useful for symmetrical distributions (normal)

• most statistics based on means

---

• Salary is not summarized with means

  • Typically positively skewed, therefore not useful

Outlier

• an extreme score

  • Often extreme high values (range from 0 to infinity)

• Only one is enough to throw the entire mean

• Common when sample sizes are smaller

Non-normal distribution - open-ended

• range from something to infinity (no limit)

  • e.g. number of followers (0 to more than millions)

• Positive skewed is likely

• Image: the intervals are inconsistent

Multiple samples, multiple means

• Multiple samples from the same population

  • May range in diff sizes

• Due to sampling error, not every mean will be the same

The default approach

• how to calculate the mean with multiple sample sizes

---

The better approach (solution)

• take the mean of all the means

• Mean = (M1 + M2 + M3)/3 = 132/3 = 41

• Issue: the sample sizes are different, and still treating each of them equallly

  • Solution: weight them according to the sample sizes

    • e.g. like grades

    • Weighted Mean = M1 20/43 + M2 19/43 + M3 * 4/43 = 34.9 - mean is more smaller (better representative)

View: The mean as a “balance point“

• calculate for example

distance of scores below mean = distance of scores above mean

• e.g. 7 below and 7 above

Delta Scores

• The deviations from the mean

• Sum of all deviations is 0 (the balance point view)

• e.g. change in attention time following training

  • Negative stimuli vs positive stimuli or change time spent after stimuli

What happens to the mean if…

• You replace a single score with one of a different value?

• You remove a score?

• You multiple ALL scores by a constant value (e.g. X * 4)?

• You divide ALL scores by a constant value (e.g. X / 5)?

• The mean changes

• The mean changes

• mean multiples by 4

• The mean divides by 5

Median

• main def

• diif strats to calculate (odd vs even)

• midpoint in a group of rank-ordered scores, separates distribution into 2 equally-sized groups

• Diff strats

  • if n = odd and no repeats, the median is the middle score

  • if n = even number, no repeats, median calculated between two scores closest to the middle

  • For cases in the same value, it is harder to find the median

---

• useful for ordinal (rank-ordered) data

• Useful for data not normal/skewed (e.g. in the image

  • e.g. salary (median is lower than the mean)

Case: Likert scales

• data is ordinal (where rank matters)

  • e.g. agreement (strong disagree, disagree, etc)

• Summing multiple items, totals more often works better with means

Finding medians

• Interpolation

• Median = Lower limit of interval + (𝑛/2 – prior cumulative f) / f class width Median = 39.5 + ((100/2 – 35)/50) 10 = 42.5

For repeating scores

• relook

Mode

• Strict definition: most common score in the distribution (than neighbouring values; highest frequency)

• Informal def: Score that is more common than nearby scores

• There can be one mode or several modes (multimodal)

  • e.g. in grades

  • unimodal vs bimodal vs multimodal

    • Bimodal is still considered symmetrical

• Best for nominal (the most common) data

1, 2, 3, 4, 4, 4, 4, 6

• find the mean, median and mode

• mean:

• Median:

• Mode: 4

• which measure would you use in each case

• first: mean

• Second: median (skewed)

• Third: bimodal

  • Means can also be used for bimodal (often an exagerating)

Normal distrubution all measures

• mean= median = mode

The case of missing data or strongly unreliable data

• what do you do? what do you replace it with

• Possible things individuals will do

  • replace with mean/media/mode/remove entirely

• Cause: maybe an error in the recording device or unusual response patterns

• reminder: to state it happened and expect to get criticized for it

• e.g. exam grades

  • removed entirely

• A median is the midpoint between the highest and lowest scores (i.e. the range of the population).

  • TRUE or FALSE

• Of the three measures of central tendency, which is the only one which MUST correspond to actual score in the data?

• A distribution has both median < mean and mode < mean. What type of distribution is it?

• A median is the midpoint between the highest and lowest scores (i.e. the range of the population).

  • TRUE or FALSE

    • The number of score - not the middle range + there may be repeats

• Of the three measures of central tendency, which is the only one which MUST correspond to actual score in the data?

  • mode (median and mean may change due to outlier = may not be representative)

• A distribution has both median < mean and mode < mean. What type of distribution is it?

  • Positively skewed

Variability

• quantitative measure of the differences between scores in a distribution and the degree to which they cluster/are spread out

• Three measures: Standard deviations, variance

• Deviation: how far score differs from the mean

• Variance: the avg of the squared deviations

• Standard diviation: the square root of the variance

Difference in sample variable and population variance

• sample variation underestimates population variance

Standard Deviation (SD)

• equation for sample and for population

• Squared: to work with positive values

• N - 1 used instead to correct the idea that the variability is going to be less than the actual population

• Mu = mean

Practice

Standard deviation - when the mean is not whole in sample SD

• what equation is used

• mean not whole is tedious to calculate

The sum of squares

• Sum of squares = the numerator of the equation for SD

• For populations:

  • Definitional: SS = (X - µ)2

  • Computational: SS = X 2 – (X) 2 N

• For samples:

  • Definitional: SS = (X - M) 2

  • Computational: SS = X 2 – (X) 2 n

Purpose of n-1

• because the sample are less variable, used as an error variance, related to degrees of freedom

Summary of SD

• A line outward from the mean • Adding a constant to all scores affects the SD in the same way • Multiplying all scores by a constant affects the SD in the same way

Why is variability bad in results?

• hard to reliably detect if there are effects

Data visualization

• mean with variability measure

Variance

• variance = SD²

• If you add a constant to all scores in the population, how do the mean and standard deviation change?

• If you multiply all scores in the population, how does the standard deviation change?

• Comment on the two following distributions:

  • M = 1000, SD = 100 vs. M = 300, SD = 100

• If you add a constant to all scores in the population, how do the mean and standard deviation change?

  • Mean changed by constant value

  • SD changes by the same amount

• If you multiply all scores in the population, how does the standard deviation change?

• Comment on the two following distributions:

  • M = 1000, SD = 100 vs. M = 300, SD = 100

    • SD is very high compared to M = less likely a normal distribution

Coefficient of variation (CV)

• CV = population standard deviation/mean

• SD as a percentage of the mean

• Higher value = the distribution is not normal

• is COV is high.= positive skew

Data transformation

• Distribution is not normals = raw scores problematics = other options

• Transform data (e.g. take the log; easier to appreciate the value)

• or alternative methods

Comparisons - how to approach potential difference in diff samples

Relative performance - comparisons in SD

• e.g. samir 76 is very high from the standard deviation (very high standing in the cohort)

• e.g. Rheanna 76 is within tht eSD (middle standing within the cohort)

Z-score method

• standardizing distributions to make scores in each comparable

• Same mean = 0 and same SD = 1

• Compare ranking

The Z score approach

• A Z score is effectively the deviation of a score from the mean in terms of SD units

• e.g. Samir is 2 SDs away, very high outlier

• e.g. Rheanna is 0.5 Sds away, not unnormal

Z score distribution

• mean = 0

• SD = 1

• Sign: + or - refers direction - matter in calculations for the z score

• Numerical value: the magnitude

Does transformation change the distribution shape

Transformation does not change distribution shape

• Does transformation change the shape of distributions?

• In a population with a mean of μ = 65, a score of X = 59 corresponds to z = −2.00. What is the standard deviation for the population?

• In a population distribution, a score of X = 54 corresponds to z = +2.00 and a score of X = 42 corresponds to z = −1.00. What are the values for the mean and the standard deviation for the population?

• Does transformation change the shape of distributions?

  • No

• In a population with a mean of μ = 65, a score of X = 59 corresponds to z = −2.00. What is the standard deviation for the population?

• In a population distribution, a score of X = 54 corresponds to z = +2.00 and a score of X = 42 corresponds to z = −1.00. What are the values for the mean and the standard deviation for the population?