Statistical Measures

MEASURES OF CENTER A measure of center is a single, central value that summarizes a set of data. The mean of a set of data values is the sum of the data divided by the number of data values. The median is a middle value when the data values are arranged in numerical order.

examples At Yellowstone National Park, Data Girl and Ms. Adventure watch Jewel Geyser erupt. Data Girl records the time intervals between eruptions. Find the mean and median intervals between eruptions. Using the measure of center, what inference can you make about how often Jewel Geyser erupts?

examples MEAN = (5.5+6+7+7+8+8.5+10+10)/8 = 62/8 = 7.75 = 7 mins. 45 secs. MEDIAN = 5.5, 6, 7, 7, 8, 8.5, 10, 10 = 7+8/2 = 15 2 = 7.5 = 7 mins. 30 secs. The mean and median are close together. The dot plot shows that the two values at 10 minutes are higher than the rest of the data, so the median may describe the more typical central value. You can infer that Jewel Geyser erupts about every 7 minutes and 30 seconds.

EXAMPLES During eruptions at Jewel Geyser, water soars up to various heights. What is the mean height of the geyser’s eruption? Heights of Jewel Geyser Eruptions (feet) 15, 30, 27, 23, 28, 19, 14, 11, 22

examples MEAN = (15+30+27+23+28+19+14+11+22)/9 = 189/9 = 21 ft

MEASURES OF VARIABILITY A measure of variability is a single value that describes the spread of values in a data set. The range of a data set is the difference between the greatest and the least values. The quartiles of a data set divide the data set into four parts with the same number of data values in each part. The interquartile range (IQR) is the difference between the first and third quartiles of the data set. It represents the spread of the middle 50% of the data values.

examples Range: IQR:

examples RANGE = 201 – 144 = 57 INTERQUARTILE RANGE (IQR) = 189 – 155 = 34 The range is 57°F and the interquartile range is 34°F. This means that the temperatures of the hot springs are spread out evenly throughout the data set.

What is the IQR of the depths of a sample of hot spring pools in Yellowstone National Park? Depths of Hot Springs (feet): 25, 6, 27, 23.5, 25, 32.5

EXAMPLES Ms. Adventure and Data Girl are thinking about their next trip. They sample the flight prices of two airlines at random. Find the median and the range for each airline. Based on the two values, make an inference about which airline they will most likely choose. Justify your reasoning.

EXAMPLES MEDIAN Beta – 399, 400, 402, 413, 722 MEDIAN = 402 Park – 398, 409, 428, 447, 465 MEDIAN = 428 RANGE Beta = 722 – 399 = 323 Park = 465 – 398 = 67 The median is not affected by stray data values, while the range is. Since there is an unusually high price in one of the samples, use the median to make your inference. The median price of the Beta Air flights is lower than the median price of Park Air flights. Ms. Adventure and Data Girl will most likely choose Beta Air.

blue, black, black, blue, green, black, green, blue, red, green, blue examples Data Girl wants to buy a new suitcase for her next trip. She wants an unusual color to make the bag easy to spot, so she records every third suitcase that comes by on the baggage claim.   blue, black, black, blue, green, black, green, blue, red, green, blue The store Data Girl shops at sells black, blue, red, and green suitcases. Which color suitcase should she buy?

examples Count the number of each color of suitcase in the sample. Blue = 4 Black = 3 Green = 3 Red = 1 Data Girl wants an unusual color, and there is only one red suitcase in her sample. She should buy a red suitcase.

Practice A and Practice B HW: Practice A and Practice B p. 309-310 in the Statistical Measure PDF