Measures of Central Tendency Mean - the arithmetic average Use to represent a population mean Use to represent a sample mean Formula: Formula: is the capital Greek letter sigma – it means to sum the values that follow parameter statistic This is on the formula sheet, so you do not have to memorize it.
Measures of Central Tendency Median - the middle of the data; 50 th percentile Observations must be in numerical order Is the middle single value if n is odd The average of the middle two values if n is even NOTE: n denotes the sample size
Measures of Central Tendency Mode – the observation that occurs the most often Can be more than one mode If all values occur only once – there is no mode Not used as often as mean & median
Range- The difference between the largest and smallest observations. This is only one number! Not 3-8 but 5 Measures of Central Tendency
Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median. 2 3 4 8 12 The numbers are in order & n is odd – so find the middle observation. The median is 4 lollipops!
Suppose we have sample of 6 customers that buy the following number of lollipops. The median is … 2 3 4 6 8 12 The numbers are in order & n is even – so find the middle two observations. The median is 5 lollipops! Now, average these two values. 5
Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean. 2 3 4 6 8 12 To find the mean number of lollipops add the observations and divide by n.
What would happen to the median & mean if the 12 lollipops were 20? 2 3 4 6 8 20 The median is... 5 The mean is... 7.17 What happened?
What would happen to the median & mean if the 20 lollipops were 50? 2 3 4 6 8 50 The median is... 5 The mean is... 12.17 What happened?
Resistant - Statistics that are not affected by outliers Is the median resistant? Is the mean resistant? Is the mean resistant? YES NO
Now find how each observation deviates from the mean. What is the sum of the deviations from the mean? Look at the following data set. Find the mean. 2223242525262930 0 Will this sum always equal zero? YES This is the deviation from the mean.
Look at the following data set. Find the mean & median. Mean = Median = 21232324252526262627 27272728303030313232 27 Create a histogram with the data. (use x-scale of 2) Then find the mean and median. 27 Look at the placement of the mean and median in this symmetrical distribution.
Look at the following data set. Find the mean & median. Mean = Median = 222928222425282125 2324232636386223 25 Create a histogram with the data. (use x-scale of 8) Then find the mean and median. 28.176 Look at the placement of the mean and median in this right skewed distribution.
Look at the following data set. Find the mean & median. Mean = Median = 214654475360555560 5658585858626364 58 Create a histogram with the data. Then find the mean and median. 54.588 Look at the placement of the mean and median in this skewed left distribution.
Recap: In a symmetrical distribution, the mean and median are equal. In a skewed distribution, the mean is pulled in the direction of the skewness. In a symmetrical distribution, you should report the mean ! In a skewed distribution, the median should be reported as the measure of center!
Quartiles Arrange the observations in increasing order and locate the median M in the ordered list of observations. The first quartile Q 1 is the median of the 1 st half of the observations The third quartile Q 3 is the median of the2nd half of the observations.
What if there is odd number? 16 19 24 25 25 33 33 34 34 median When dividing data in half, forget about the middle number
The interquartile range (IQR) The distance between the first and third quartiles. IQR = Q 3 – Q 1 Always positive
Outlier: We call an observation an outlier if it falls more than 1.5 x IQR above the third or below the first. Lets look back at the same data: 16 19 24 25 25 33 33 34 34 37 37 40 42 46 49 73 Q 1 =25Q 3 =41 IQR=41-25=16 25 - 1.5 x 16 = 141 + 1.5 x 16 = 65 Lower Cutoff Upper Cutoff
Since 73 is above the upper cutoff, we will call it an outlier.
Five-number summary Minimum Q 1 Median Q 3 Maximum
If you plot these five numbers on a graph, we have a ………
Advantage boxplots? ease of construction convenient handling of outliers construction is not subjective (like histograms) Used with medium or large size data sets (n > 10) useful for comparative displays
Disadvantage of boxplots does not retain the individual observations should not be used with small data sets (n < 10)
How to construct find five-number summary Min Q1 Med Q3 Max draw box from Q1 to Q3 draw median as center line in the box extend whiskers to min & max
Modified boxplots display outliers fences mark off the outliers whiskers extend to largest (smallest) data value inside the fence ALWAYS use modified boxplots in this class!!!
Modified Boxplot Q1 – 1.5IQRQ3 + 1.5IQR Any observation outside this fence is an outlier! Put a dot for the outliers. Interquartile Range (IQR) – is the range (length) of the box Q3 - Q1 These are called the fences and should not be seen.
Modified Boxplot... Draw the whisker from the quartiles to the observation that is within the fence!
A report from the U.S. Department of Justice gave the following percent increase in federal prison populations in 20 northeastern & mid-western states in 1999. 5.9 1.35.05.94.188.8.131.52.86.9 4.5 184.108.40.206.55.38.04.47.23.2 Create a modified boxplot. Describe the distribution. Use the calculator to create a modified boxplot.
Evidence suggests that a high indoor radon concentration might be linked to the development of childhood cancers. The data that follows is the radon concentration in two different samples of houses. The first sample consisted of houses in which a child was diagnosed with cancer. Houses in the second sample had no recorded cases of childhood cancer. (see data on note page) Create parallel boxplots. Compare the distributions.
Cancer No Cancer 100 200 Radon The median radon concentration for the no cancer group is lower than the median for the cancer group. The range of the cancer group is larger than the range for the no cancer group. Both distributions are skewed right. The cancer group has outliers at 39, 45, 57, and 210. The no cancer group has outliers at 55 and 85.
Linear transformation rule When multiplying or adding a constant to a random variable, the mean and median changes by both. When multiplying or adding a constant to a random variable, the standard deviation changes only by multiplication. Formulas:
An appliance repair shop charges a $30 service call to go to a home for a repair. It also charges $25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor?
Rules for Combining two variables To find the mean for the sum (or difference), add (or subtract) the two means To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root. Formulas: If variables are independent
Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times? PhaseMeanSD Unpacking3.50.7 Assembly21.82.4 Tuning12.32.7