1. Intro to Descriptive Statistics
GTECH 201 Lecture 12

2. Topics for Today Measures of Central Tendency Measures of Dispersion
Mean, Median, Mode
Sample and Population Mean
Weighted Means
Selecting Appropriate Measures of Central Tendency
Measures of Dispersion
Variance
Standard Deviation

3. Descriptive vs. Inferential
Descriptive Statistics
Methods for organizing and summarizing information
Inferential Statistics
Methods for drawing and measuring the reliability of conclusions about a population based on information obtained from a sample of the population

4. Looking at This Data Set…
Student Performance in Class Tests
Observations: 8
Examples of Variables: Test 1, Test 2
Data Value: for Observation 1 in Test 2, Data Value = A
Variables – can have one data value

5. Overview Mean Median Mode Sample and Population Mean Weighted Means
Selecting Appropriate Measures of Central Tendency
Applying these measures

6. Mean The mean of a set of n observations is the arithmetic average
Mean of n observations x1, x2,x3,….xn is
In Excel, =AVERAGE(insert range)

7. Median
The data value that is exactly in the middle of an ordered list if the number of pieces of data is odd
The mean of the two middle pieces of data in an ordered list if the number of pieces of data is even
The median is a typical value; it is the midpoint of observations when they are arranged in an ascending or descending order

8. Mode
The most frequent data value; i.e., any value having the highest frequency among the observations
In Excel,you use the functions
=MEDIAN (insert range)
=MODE (insert range)
Unimodal, Bimodal, Multimodal data sets
Outliers

9. Sample and Population Means
Mean of a data set
Population mean if data set includes entire population
Sample mean if data set is only a sample of the population

10. Weighted Means
To calculate the mean when your information is available only in the form of summary data
C Interval Freq
25 –
30 –
35 –

11. Skewed Distributions

12. Skewed Distributions
When there is one mode and the distribution is symmetric
mean, median, mode are the same
Positive skew
mean moves towards the positive tail
median also pulls towards the positive tail
Negative skew
mean moves towards the negative tail
median also moves towards the negative tail

13. Selecting Appropriate Measures
Mean
affected by extreme values
includes all observations, therefore comprehensive (useful for interval/ratio data)
Median
not affected by the number of observations
reveals typical situations (used for ordinal data)
Mode
useful for nominal variables

14. Other Useful Calculations
In addition to the sum of data, Sx we need to be able to calculate:

15. Variability or Spread Mean and the median - limits
Range – coarse measure of variability
Percentiles
kth percentile is the point at which k percent of the numbers fall below it and the rest are fall above it
25th percentile (lower quartile)
50th percentile (median)
75th percentile (upper quartile)
Interquartile range (difference between the 25th percentile value and the 75th percentile value)

16. Describing the Spread A five number summary
Median
Quartiles
Extremes
Variance and Standard Deviation
Measures spread about the mean
Standard deviation cannot be discussed without the mean

17. Calculating Percentiles
In the list of twelve observations
Compute median, 25th and 75th percentiles
The lower quartile is the median of the 6 observations that fall below the median
The upper quartile is the median of the 6 observations that fall above the median

18. Five Number Summary Median = 11 Lower Quartile = 9 Upper Quartile = 16
Extremes are 2 and 29
Can compute the range = 27
In a symmetric distribution, the lower and upper quartiles are equally distant from the median

19. Variance
Is the mean of the squares of the deviations of the observations from their mean
Population variance
Sample variance

20. Example
The heights, in inches for five starting players in a men’s college basket ball team are:
Compute the mean and standard deviation.
= 75

21. Standard Deviation
Standard deviation is positive square root of the variance
Variance in our basketball example:
= 39

22. Formulas – Standard Deviation
Standard deviation of a sample
Standard deviation of a population

23. Example (Continued)

24. Short Cut – Simpler Formula
Standard Deviation
of a sample
Sum of the squares of data values,
i.e., you square each data value and then sum those squared values
Square of the sum of data values,
i.e., you sum all the data values and then square that sum

25. Example (using the short cut)

26. Interpreting Std. Deviation
s and s 2 will be small when all the data are close together
The deviations from the mean
Will be both positive and negative
Sum will always be 0
s is always 0 or a positive number
s = 0 means no spread; as s value increases, the spread of the data increases
The units of s are the same as the original observations
s is heavily influenced by outliers

27. Coefficient of Variation
CV is the standard deviation described as a percent of the mean
CV =
CV is useful when comparing different sets of data where sample size and standard deviation are different