Introduction
Overview of Descriptive Measures
Measures of Central Tendency
Mean
· A mean is the most common measure of central tendency.
- A mean is what we commonly think of as the ‘average’ value.
- Calculate the mean by summing the values and dividing by the total number of values.
90+80+70+85+70 = 395
Mean = 395/5 = 79
· Extremely large values in a data set will increase the value of the mean, and extremely low values will decrease it.
- To calculate a weighted mean, first, multiply each cell frequency by its weight or by the cell frequency, and then sum and divide by the total frequency.
| X | f | f * X | |
| 90 | 1 | 1*90 = | 90 |
| 80 | 2 | 2*80 = | 160 |
| 70 | 3 | 3*70 = | 210 |
| 85 | 2 | 2*85 = | 170 |
| 70 | 1 | 1*70 = | 70 |
| 700 | |||
Mean = 700/5 = 140
Median
· The median is the central point of the data.
· Half of the data has a lower numerical value than the median.
· Half of the data has a higher numerical value than the median.
· The median is not affected by extremely large or small values.
· To find the median, arrange the data in order from smallest value to largest value, and
Þ if there are an odd number of points, find the value that is in the center of the data
Þ if there are an even number of points, add the two middle values and divide by 2.
For the values (written in order): 70 70 80 85 90
the median is 80.
Mode
- The mode is the data value that occurs most frequently.
- The mode is not affected by extreme values.
For the values: 70 70 80 85 90
the mode is 70.
Measures of Spread
Range
· Subtract the smallest value from the largest – or
· Report the smallest and largest values.
Variance/Standard Deviation
· The standard deviation is the average variation of the data values from the mean of the values.
· The standard deviation is found by taking the square root of the variance, and the standard deviation is more useful than the variance in reporting results so it is the measure that is typically reported.
| X | X – Mean | (X – Mean)2 | ||
| 90 | 90-79 = | 11 | 11*11 = | 121 |
| 80 | 80-79 = | 1 | 1*1 = | 1 |
| 70 | 70-79 = | -9 | -9*-9 = | 81 |
| 85 | 85-79 = | 6 | 6*6 = | 36 |
| 70 | 70-79 = | -9 | -9*-9 = | 81 |
| 395 | 320 | |||
| Mean = 395/5 = 79 | ||||
| Sample Variance = 320/4 = | 80 | |||
| Sample Standard Deviation = | 8.9 | |||
| Population Variance = 320/5 = | 64 | |||
| Population Standard Deviation = | 8.0 | |||
The Empirical Rule
· Apply this rule to interpret the measures when the data is symmetrical.
· At least:
68% of the data values are within one standard deviation of the mean
90% of the data values are within two standard deviations of the mean
99% of the data values are within three standards deviation of the mean
Tchybychef’s Inequality
· Apply this method to interpret the measures when the data is skewed.
· At least:
75% of the data values are within two standard deviations of the mean.
90% of the data values are within three standard deviations of the mean.
Measures of Relative Standing
Percentiles
If your percentile score on the GRE is 90 then you scored better than 90% of those taking the test, and you scored lower than 10% of those taking the test,
Quartiles
· The lower quartile is the same as the 25th percentile.
Þ 25% of the scores are lower and
Þ 75% of the scores are higher than the lower quartile.
· The upper quartile is the same as the 75th percentile.
Þ 75% of the scores are lower and
- 25% of the values are greater than the upper quartile.
Correlation
- Correlation is used in describing the strength of the relationship between two (or more) variables.
- There are many different types of correlation coefficients and selection of the appropriate one depends on the form of the variables. We will consider Pearson Product-moment Correlation Coefficient which assumes continuous quantitative data.
- Correlation coefficients reflect whether the relationship between variables is:
- positive (i.e. as one variable increases, the other variable increases) or
- negative (i.e. as one variable increases, the other variable decreases).
It also may indicate that there is no relationship.
- Borg and Gall, Educational Research from Longman Publishing, provide the following information for interpreting correlation coefficients.
- Correlation coefficients ranging from 0.20 to 0.35 show a slight relationship between the variables; they are of little value in practical prediction situations.
- With correlation around 0.50, crude group prediction may be achieved. In describing the relationship between two variables, correlation coefficients that are this low do not suggest a good relationship.
- Correlation coefficients ranging from 0.65 to 0.85 make possible group predictions that are accurate enough for most purposes. Near the top of this correlation range, individual predictions can be made that are more accurate than would occur if no such selection procedure were used.
- Correlation coefficients over 0.85 indicate a close relationship between the two variables.
- It is important to understand that even a high correlation coefficient does not establish a cause and effect relationship. There may be other factors that relate to both of the variables.
- In comparing two variables, you can take the square root of the correlation coefficient to get the Coefficient of Determination; this measure gives the percent of the variation in the dependent variable that is ‘explained’ by the independent variable.
- It is always good to look at an XY scatter plot to see what you think about the relationship between the variables.
- Excel will not only give you a correlation coefficient, but it will also give you the equation for the Least Square line which can be useful in describing the relationship between the two variables and in making predictions of the dependent variable from the independent variable.
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