Psych 300A: Midterm 1 Review (Central Tendency & Variability)

Central Tendency

A statistical measure to determine a single score that defines the centre of a distribution

What are the 3 measures of central tedency

1. Mode

2. Median

3. Mean

Mode

The most frequent category/score in a distribution (often the high point in a graph)

In what scales of measurement can mode be used with

Nominal, ordinal and ratio/interval scale (Only scale that can work with nominal)

T or F: You can only have one mode in each dataset

F, there can be multiple if two values have the same number of frequencies

What are the 4 major advantages of the mode

1. Easy to compute/determine

2. Value that is observed in dataset (real number from data)

3. Can be used for all scales of measurement

4. Not affected by outliers

What are the 2 major disadvantages of the mode

1. Not used in most statistical computations as it is not useful for making inferences

2. May not be representative of the entire collection of numbers

What is the measure of variability with the mode

There is none as the variable cannot be quantified

Median

Physical middle of an ordered set of data, also known as the 50th percentile, can be used with ordinal and ratio/interval scale

When is the most common use for the median

When data is extremely skewed

What are the 3 major advantages of the median

1. Can be computed on data that is ordinal, interval or ratio and is the best measure of central tendency in those ranks

2. Less biased measure of central tendency when interval/ratio data is skewed

3. Not affected by outliers or extreme scores

What are the 2 major disadvantages of the median

1. Not used in statistical computations/can not be used to make inferences

2. Subject to sampling variation (not stable from sample to sample so we cannot infer anything about population)

What is the measure of variability used with the median

Find the min/max values as well as the range (max value - min value)

Mean

The average value in a dataset, can work with an interval/ratio scale

What is the symbol used for the population mean

Mu (μ)

What is the symbol used for the sample mean

M or x-bar (x̄)

What are the 3 major characteristics of the mean

1. Changing the value of any score or changing the number of scores will change the mean

2. Adding or subtracting a constant for each score in a distribution will add or subtract that same constant from the mean

3. Multiplying or dividing every score in a distribution by a constant will multiply or divide the mean by that same constant

What are the 4 major advantages of the mean

1. Can be manipulated algebraically

2. Takes into account quantitative info about each value

3. Value has the most meaning for interpretation, especially with ratio values

4. Most common value for inferential stats

What are the 3 major disadvantages of the mean

1. Applies only to interval/ratio data

2. Influenced by outliers

3. Computed value may not reflect any actual value in the dataset

What measure(s) of CT can be found with nominal data

Only the mode

What measure(s) of CT can be found with ordinal data

Median or mode, median is usually the best pick

Variability

Quantitative distance, the measure of the differences between the scores in a distribution

What are the 5 major measures of variability

1. Range

2. Semi-interquartile Range (SIQR)

3. Median Absolute Deviation (MAD)

4. Variance

5. Standard Deviation

Range

Distance coved by scores in a distribution from smallest to largest

T or F: Range is calculated differently depending on if the variables are continuous or discrete

T, Discrete values are simply max - min, whereas continuous values are the upper limit of max - upper limit of min

What are the major advantages and disadvantages of range

Advantages: Quick to compute and includes entire distribution of data

Disadvantages: Derived from only two values so spread of data is unknown and sensitive to outliers

Semi-interquartile Range (SIQR)

Half of the range of the middle 50% of observations, calculated by the (third quartile - first quartile) / 2

First quartile

25th percentile, range of the first 25% of data

Third quartile

75th percentile, range between 50% and 75% of data

Interquartile range

third quartile - first quartile

Median Absolute Deviation (MAD)

Absolute measure of how many physical units values deviate from the median, calculated as: MAD = Mdn|X - Mdn| (take the median of the each value - the median of the original data set)

What are the 3 major advantages of the MAD

1. Takes all scores into account

2. Less sensitive than SD to extreme scores or skews in data

3. It is a minimum, we cannot get smaller value even if we take the absolute deviation from another location in the dataset

What are the 2 major disadvantages of the MAD

1. Provides limited description of variability

2. Not useful in advanced statistical procedure

Variance

Average squared distance from the mean, one can calculate it by using the formula: V = SD² = (Σ(x - x̄)²) / N

What is the Sum of Squared Deviations (SS)

Sum of the squared difference between each score and its mean, calculated by: SS = ∑(x −x̄)²

Standard Deviation

Measure of standard/average distance from the mean (how dispersed scores are around the mean), calculated: SD = √SD² = √variance

What measure of variability can by used with a nominal scale

None, needs quantitative values

What measure of variability can by used with a ordinal scale

Range, SIQR, MAD

What measure of variability can by used with a interval/ratio scale

Range, SIQR, MAD, Variance, SD

What are the two characteristics of standard deviation

1. Adding or subtracting a constant to every score in a distribution will not change the standard deviation

2. Multiplying or dividing every score in a distribution by a constant will multiply or divide the standard deviation by the same constant

What are the 3 advantages of SD

1. Accounts for all scores in a distribution

2. Good description of variability

3. Used in many advanced stats

What are the 2 disadvantages of SD

1. Can only be used with interval/ratio data

2. Sensitive to extreme scores or outliers and is biased when distributions are skewed

Pearson coefficient of skew (Skew_p)

Helps us understand the magnitude of skew, calculated using a skew statistic (in our case the pearson coefficient of skew)

How does one calculate Skew_p

Skew_p = 3(x̄ - Mdn) / SD

What does the sign of Skew_p tell us

The direction of skew (positive or negative)

What does the magnitude Skew_p tell us

The degree of skew, no skew and Skew_p = 0

What values of Skew_p demonstrate a normal distribution and what measure of variability should be used

Between 0 and |.5|, use mean and SD

What values of Skew_p demonstrate a mild/moderate skew and what measure of variability should be used

Between |.5| and |1.0|, use mean and SD

What values of Skew_p demonstrate a moderate/strong skew and what measure of variability should be used

Between |1.0| and |2.0|, use mean and SD if ≤ |1.5|, else use median and MAD

What values of Skew_p demonstrate a severe skew and what measure of variability should be used

Greater than |2.0|, use the median and MAD

T or F: We do not need central tendency and variability to fully grasp the shape of a distribution

F, we need both

Explain what each variable in the formula represents:

X₀= X_t ± X_e

X₀: represents our observed value

X_t: represents our measure of central tendency

X_e: represents our measure of error

Why is measure of error so important

Allows us to make hypothesis on why something occured

Anscombes quartet

4 datasets with nearly identical summary stats, however the actual datasets are different which can be demonstrated with graphs

What is the overall conclusion we can make about shape, central tendency, variability and data observation

We need all of them if we are to be able to make assumptions about our data