Percentiles, Interpolation, Stem-and-Leaf Displays, and Measures of Central Tendency

Percentiles, Cumulative Frequencies, and Interpolation (Lecture 12)

• Percentile Rank: This is defined as the percentage of individuals in a distribution with scores at or below a particular value.

• Percentile: When a specific score is identified by its percentile rank, that score is referred to as a percentile. Whereas "percentile rank" refers to a percentage, "percentile" refers to the actual value or score.

• Cumulative Frequencies ($cf$):

  • Definition: These represent the accumulation of individuals as you move up the measurement scale.

  • Computation: To compute the cumulative frequency for any row in a frequency distribution table, you must add up the frequencies in and below that specific value.

  • Identification: In a frequency distribution table, these values are found in the column labeled "cf".

• Cumulative Percentages ($c\%$):

  • Definition: The percentage of individuals located at or below each specific value in the distribution.

  • Computation: Calculate by taking the cumulative frequency ($cf$) for a row, dividing it by the total number of scores ($N$), and multiplying by 100: $\text{Cumulative Percentage} = (\frac{cf}{N}) \times 100$.

  • Identification: These are found in the column labeled "c%".

  • Relationship to Real Limits: Each cumulative percentage value is associated with the upper real limit of its corresponding interval.

• Determining Percentiles and Percentile Ranks:

  • Direct Determination: Percentiles and ranks can be determined directly if the specific percentile ranks (percentages) appear explicitly in the table.

  • Interpolation: If a value does not appear in the table, a procedure called interpolation is used to obtain estimates of these intermediate values.

• The Interpolation Process: This is a method for finding values located between two specified numbers to estimate intermediate values not explicitly given in a table. The general process is as follows:

  1. Find the width of the interval on both scales (e.g., the score scale and the percentage scale).

  2. Locate the position of the intermediate value as a fraction of the whole interval: $\text{fraction} = \frac{\text{distance from the top of the interval}}{\text{interval width}}$.

  3. Use this fraction to find the distance from the top on the alternative scale: $\text{distance} = \text{fraction} \times \text{width}$.

  4. Use that distance calculation to determine the exact position on the second scale.

• RStudio: Cumulative Frequencies and Percentages:

  • cumsum() function: This is used to calculate cumulative frequencies. It takes a frequency distribution as input and returns a vector containing the cumulative frequencies.

  • Procedure for Adding to a Table:

    1. Calculate cumulative frequencies using cumsum().

    2. Create a data frame using the rev() function on both the frequency table and the cumulative frequencies. This is necessary because table() and cumsum() produce values from low to high; rev() ensures they are presented correctly.

    3. Calculate the total number of scores ($N$) by summing the frequencies via sum().

    4. Use the cbind() function to add the cumulative percentages column where each value is $\frac{cf}{N} \times 100$.

    5. Fix column names with colnames() and print the result using kable().

Stem and Leaf Displays (Lecture 12 Continued)

• Stem and Leaf Display: A simple alternative to a frequency distribution table or graph.

• Stem: The first digit or digits of a score.

• Leaf: The last digit or digits of a score.

• Construction: List all stems in a vertical column and go through the data set, writing the leaf for each score beside its corresponding stem.

• Advantages:

  • It is easy to construct.

  • It identifies every individual score in the data set.

  • It provides both a complete listing of scores and a visual "picture" of the distribution.

• Splitting Stems: In some cases, stems are split to provide a more detailed picture. This involves associating the first half of each stem with lower leaves ($0$-$4$) and the second half with upper leaves ($5$-$9$).

• The stem() function in RStudio:

  • Purpose: Creates a stem and leaf display.

  • Ordered Leaves: The function automatically displays leaves in order, which helps in identifying specific values of interest.

  • Lowest Score: Identified as the first leaf of the first stem.

  • Highest Score: Identified as the last leaf of the last stem.

  • Most Common Score: Found by scanning for large groups of the same number within a stem.

  • Count in Range: One can look at specific stems and leaves to count how many subjects scored below a certain value or between two values.

  • scale parameter: This parameter adjusts the height of the display. For example, scale=2 doubles the number of stems (splitting them).

Measures of Central Tendency: The Mean (Lecture 13)

• Measure of Central Tendency: A statistic that describes a distribution by identifying a single score as most representative or typical of the entire distribution (the "center").

• Three Primary Measures: The mean, the median, and the mode.

• The Mean (Average):

  • Formula for Populations: $\mu = \frac{\Sigma X}{N}$.

  • Formula for Samples: $\bar{X} = \frac{\Sigma X}{n}$.

  • Notation: Use the Greek letter mu ($\mu$) for the population mean and X-bar ($\bar{X}$) for the sample mean.

  • Computation: Sum all scores ($\Sigma X$) and divide by the total number of scores ($N$ or $n$).

• Weighted Mean: The overall mean for a combined group of scores from two or more separate groups.

  • Calculation: Combine the sum of scores ($\Sigma X$) from all groups and divide by the combined sample sizes ($n$) of all groups.

  • RStudio Methods:

    1. If given specific means and n-values: $\text{Weighted Mean} = \frac{\text{Mean}_1 \times n_1 + \text{Mean}_2 \times n_2 + \dots}{n_1 + n_2 + \dots}$.

    2. If given raw scores: Use sum() to find $\Sigma X$ and length() for $n$ of each group, then divide the total sum by the total length.

    3. Proportional Weights: Multiply each group mean by its proportional weight (percentage of the total group) and sum the results.

• Characteristics of the Mean:

  • Changing a Score: Changing any score, adding a score, or removing a score will change the mean because it alters the sum ($\Sigma X$) and/or the count ($n$).

  • Adding/Subtracting a Constant: If a constant is added to or subtracted from every score in a distribution, the mean will change by that same constant.

  • Multiplying/Dividing by a Constant: If every score is multiplied or divided by a constant, the mean will be multiplied or divided by that same constant.

• mean() function: Returns the mean of a set of scores when passed a vector or data frame column.

The Median and Mode (Lecture 13 Continued)

• The Median:

  • Definition: The score that divides a distribution exactly in half, such that $50\%$ of the scores are at or below it.

  • Equivalent Term: The 50th percentile.

  • Notation: There are no special symbols; it is identified by the word "median".

• Calculating the Median:

  • If N is Odd: List scores in order; the median is the middle score in the list.

  • If N is Even: List scores in order; the median is the point halfway between the two middle scores.

  • Tied Middle Scores: If there are tied scores in the middle of the distribution, interpolation is used to find the precise midpoint.

• RStudio: Median Functions:

  • median() function: Takes a vector of raw scores and returns the middle number (if $N$ is odd) or the average of the middle two (if $N$ is even). Importantly, this function does not use interpolation for tied scores.

  • interp.median() function: Used to find the interpolated median. This function is part of the "psych" package.

• The Mode:

  • Definition: The most common observation or the category with the greatest frequency.

  • Determination: Identify the score located directly beneath the highest point in a frequency distribution.

  • Major vs. Minor Mode: In distributions with two unequal peaks, the tallest peak is the major mode, and the shorter peak is the minor mode.

  • Classification:

    • Unimodal: A distribution with one mode.

    • Bimodal: A distribution with two modes.

    • Multimodal: A distribution with more than two modes.

    • Amodal: A distribution with no mode (all points have equal frequency).

• Finding the Mode in RStudio:

  1. Call table() to tabulate the frequency distribution.

  2. Use sort() with the argument decreasing=TRUE to place the most frequent value at the first position.

  3. Use the names() function on the first position ([1]) of that sorted table to report the mode value.

Selecting a Measure of Central Tendency and Distribution Shape (Lecture 14)

• Preferred Measure: The mean is generally the preferred measure of central tendency because it is influenced by every single score in the distribution and is closely related to measures of variability (variance and standard deviation).

• When to use the Mode:

  • Advantages: It is easy to identify at a glance and can be used with any scale of measurement.

  • Nominal Scales: The mode is the only way to describe central tendency for nominal data, as the mean and median are mathematically impossible or meaningless for non-numerical categories.

• When to use the Median (Four Situations):

  1. Extreme Scores: Scores very different from others in the distribution. The median is not easily affected by these, whereas the mean is.

  2. Undetermined Values: Occurs when a subject has an unknown score (e.g., failing to finish a task). The median can be used because scores can still be ranked, but the mean cannot be calculated because $\Sigma X$ is unknown.

  3. Open-Ended Distributions: When there is no upper or lower limit for a category (e.g., "$5$ or more"). The median finds the midpoint, but the mean cannot be calculated.

  4. Ordinal Scale: The median is the preferred and appropriate measure for ordinal data.

• Distribution Shape and Central Tendency:

  • Symmetrical Distributions: The mean and median will be identical and located at the center.

    • Unimodal: Mean, median, and mode all share the same value at the center.

    • Bimodal: Mean and median are at the center; modes are located on either side of the center.

    • Rectangular: No mode exists, but the mean and median remain at the center.

  • Positively Skewed: The peak (mode) is on the left. The median is to the right of the mode, and the mean is to the right of the median (\text{Mean} > \text{Median}).

  • Negatively Skewed: The peak (mode) is on the right. The median is to the left of the mode, and the mean is to the left of the median (\text{Mean} < \text{Median}).

• Identifying Shape from Values:

  • If $\text{Mean} = \text{Median}$, the distribution is symmetrical.

  • If \text{Mean} > \text{Median}, the distribution is positively skewed.

  • If \text{Mean} < \text{Median}, the distribution is negatively skewed.