Summary Statistics: Measures of Location and Dispersion

The sum of values,, can be denoted as.

 Select 4 students and ask “how many brothers and sisters do you have?” Data: 2, 3, 1, 3 Or we can write

 Solve the following:

 Measure of Central Tendency - Description of Average (Typical Value)  Sample Mean:

 number of siblings – Data: 2, 3, 1, 3  Suppose we had selected a 5 th person for our sample which had 10 siblings. New Data: 2, 3, 1, 3, 10  The sample mean is sensitive to extreme values and does not have to be a possible data value.

 rank data from smallest to largest  if n is odd, median is the middle score  if n is even, median is the mean of two middle scores

 number of siblings – Data: 2, 3, 1, 3  New Data: 2, 3, 1, 3, 10  Sample median is not sensitive to extreme scores  Half the data will fall above the sample median and half below the sample median

 The median is a better measure of central tendency if extreme scores exist.  If extreme scores are unlikely, the mean varies less from sample to sample than the median and is a better measure.

 If the distribution is right skewed  If the distribution is symmetric  If the distribution is left skewed

 sample mode: most frequent score  Example: number of siblings – Data: 2,3,1,3 Mode = 3  New Data: 2,3,1,3,10 Mode = 3  Mode does not always exist/can be more than one  Also, it is unstable  Should be used with qualitative data

 Example: number of siblings – Data: 2,3,1,3  Midrange =  New Data: 2,3,1,3,10  Midrange =  Midrange is totally dependent on extreme scores.

 Percentiles – gives the percentage below an observation  Quartiles – divide the data into four equally sized parts , First Quartile: 25 th percentile , Second Quartile ( ), 50 th percentile , Third Quartile, 75 th percentile

 Order the data from smallest to largest  Find. This is  is the median of the lower half of the data; that is, it is the median of the data falling below (not including )  is the median of the upper half of the data; (same as above)

 Interquartile range (IQR) = Q3 – Q1  Range of the middle 50% of the data  5 number summary – The low score, Q1, Q2, Q3, and the high score

StudentsFaculty

StudentsFaculty Low = 0Low = 10 Q1 = 1Q1 = 15 Q2 = 5Q2 = 25 Q3 = 7Q3 = 31 High = 10High = 73

 The box goes from Q1 to Q3 and represents IQR  The line through the box is Q2 ( )  Extreme values are identified by *’s  Lines, called whiskers, run from Q1 to the lowest value and from Q3 to the highest value (If the low or high are extreme then the whisker goes to the next value)

Distribution #1Distribution # Distribution #1Distribution #2 = 35 = 35 mode = 35 midrange =35midrange = 35

Example: Years of experience of faculty Data: 1, 30, 22, 10, 5  Range is sensitive to extreme scores (Based entirely on the high and low)  Range is easy to compute

 Large values of suggest large variability  It is difficult to interpret since it is in square units  Keep in mind it can never be negative

Example: Years of experience of faculty Data: 1, 30, 22, 10, 5 sample standard deviation – measures the average distance data points are from Standard deviation is in the same units as the data

Z-score – Gives the number of standard deviations an observation is above or below the mean Example: Test scores = 79, s = 9 If your score is 88%, what is your z-score? If your score is 63%, what is your z-score?

 Approximately 68% of the data fall within 1 standard deviation of the mean  Approximately 95% of the data fall within 2 standard deviations of the mean  Approximately 99.7% of the data fall within 3 standard deviations of the mean

Example: Suppose that the amount of liquid in “12 oz.” Pepsi cans is a mound shaped distribution with oz. and s = 0.1 oz.