Chapter 3-4 statistics for social work

Measures of Central Tendency

- They are not interchangeable as they have specific meanings - Is a way to summarize the distribution of a variable within a data set.

Variability:

- to provide a more accurate description of the distribution of the variable. - Is a summary descriptor. - It provides an indicator of the degree of variation among values or value categories that occurred. It is also known as dispersion.

MEDIAN

If data can be formed into an array (ordinal level), the median can be used to report central tendency The median divides an array of values into 2 equal halves; it is a value above and below which half the values in an array fall. In other words: Is the score in the distribution that marks the 50th percentile or the midpoint but not always

MEAN

Is the most easily understood, best known and most useful of the 3 measures of central tendency Is the average (sum of all the values in a distribution divided by the total number of values) Can be computed for any interval level or ratio level variable It cannot be computed with nominal level variables or ordinal level variables

MODE

Is the value within a data set that occurs most frequently (unimodal), is the most common attribute Is considered the most unrestricted (has the fewest requirements for its use) Is not used as often as the other measures of central tendency, as it frequently does not do a good job of describing what is typical in the distribution of a variable Example, When data are at the ordinal, interval, or ratio level of measurement, can usually obtain more accurate and representative descriptions by using one or both of the other 2 measures of central tendency

Interquartile Range

Is used to handle the outliers Does not use every case value in the final calculations, which does distort the picture of the data set

Which Measure of Central Tendency to Use?

Nominal level variables---the mode should be used Ordinal level measurement—report where the median of distribution fell

Range:

a. Like the mean, the range is easily distorted by the presence of outliers. One outlier at either end of an array can greatly increase the range of a data set and suggest much more variability than is actually present. b. Vulnerability of the range to the influence of outliers is an undesirable characteristic, especially when comparing the ranges of 2 distributions of the variable. c. The presence of outlier in one distribution and not in the other can give a misleading impression about the degree of dissimilarity of the 2 distributions.

Outlier

atypical value: lies outside the area where most of the other values are found. It affects the median the least. In this case, the outlier is seen as the highest value in the data set.

MEDIAN

can be a whole number, a fraction or decimal or a mixed number that coincides with no actual case value (when have a 0 near the center and others with a frequency grater than 1 that occur near the center of the array) is a useful statistic to examine when the scores in a distribution are skewed or when there are a few extreme scores at the high end or the low end of the distribution Is affected the least by the presence of outliers

Variability (Interquartile Range)

can be reported as the range of values in an array that fall between the 75th and 25th percentiles—the range for the middle 50% Each value in a frequency distribution has its corresponding percentile Is complicated to figure out, there is statistical software that does this Is a more stable measure of variability than the range, can think of it as a kind of "trimmed range" Outliers can't distort it There is a semi-interquartile range, it is the interquartile range divided by 2, or one-half of the range between the 75th and 25th percentiles in an array of values

Interval or ratio level measures of variability

can describe the dirtribution of values

The Weighted Mean

computing an average for values that are not equally weighted (or of equal importance). Entails the "weighting" of numbers in order to arrive at a value that is more meaningful for the data set than either the arithmetic mean or the trimmed mean. A weighted mean rather than the arithmetic mean should be used in those situations where not all measurements are of equal weight or importance. It allows for fairer comparisons.

The Trimmed Mean

designed to minimize the effect of a few extreme outliers. Combines the best features of both the mean and the median. It uses most of the actual case values in its computation, but, like the median, it allows extreme values on either end to cancel each other out. Example, Done by throwing out the top 5% and the bottom 5%, then, the remaining 90% of values are averaged. This is the trimmed mean.

Measures of Variability, Measures of central tendency

tell us about the distribution of the values of a variable but cannot tell us if most values tend to cluster around a typical value or how widely they vary from the typical value.

Range:

the distance that encompasses all values within a data set. Expressed as a formula: Range= maximum value - minimum value + 1 Example: age range: 30,31,32, 33, 34, 35 35 - 30 + 1= 6

The Weighted Mean

to compute final grades of students when averaging scores on assignments and exams that are given different weights

use measures of central tendency

to provide an accurate mental image of the distribution of variables within our data set. In some situations, no single measure of central tendency would accurately represent what the data really looks like so report more than one measure of central tendency

MEAN

uses all the values within a data set in its computation. This promotes either accuracy or distortion when it is used as a measure of central tendency, depending on the degree to which a distribution of values is symmetrical. Why? If values are not symmetrical they are outliers and they can distort particularly it is a small data set but in a large data set outliers cause less distortion..

When the variable is at the ordinal or nonminal of measuremet

variability can be communicated best in frequency distribution or graph e.g. bar chart -use graph

Bimodal

when there are 2 numbers that occur frequently in a data set example: When you have more than one value occur more frequently, if a histogram was drawn it would have two peaks, when this situation occurs, you report both values as the mode for the data set and describe the distribution of the variable