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Chapter 8: Statistics: An Introduction
8.2 Measures of Central Tendency and Variation
8.2.1. Vocabulary
8.2.1.1. measures of central tendency – describe where the data are centered
8.2.1.2. measures of center – another way of describing measures of central tendency
8.2.1.3. arithmetic mean – same as the mean
8.2.1.4. average – same as the mean
8.2.1.5. mean – the sum of all data points divided by the number of data points
8.2.1.6. median – value exactly in the middle of an ordered set of numbers; the 2nd
quartile or Q2
8.2.1.7. mode – number that appears most frequently in a set; can be zero, 1, 2, 3, or
more of these
8.2.1.8. bimodal – when there are two modes
8.2.1.9. range –
highest – lowest = range: H – L = R
8.2.1.10. lower quartile – 1st quartile – Q1: The median value of the values below Q2
8.2.1.11. upper quartile – 3rd quartile – Q3: The median value of the values above Q2
8.2.1.12. interquartile range (IQR) –
Q3 – Q1 = IQR
8.2.1.13. box-and-whisker-plot – based on range and quartiles to summarize data
8.2.1.14. outlier – data point outside the expected range
n
8.2.1.15. variance –
 xi
 x
n
2
i 1
n
 s 2 or
 xi2  nx 2
i 1
n
n
8.2.1.16. standard deviation –
s  s2 
 x
i 1
i
 x
2
n
8.2.2. Computing means
8.2.2.1. Definition of Mean: The arithmetic mean of the numbers x1, x2, … , xn, denoted
x  x 2  x 3    xn
by x and read “x bar,” is given by x  1
n
8.2.2.1.1. most widely used measure of central tendency
8.2.3. Understanding the mean as a balance point
8.2.3.1. mean = balance point for data
8.2.3.2. mean does not have to be a data point and often is not
8.2.3.3. Pennies and a ruler: class demonstration
8.2.3.4. Now try this 8-4 p. 527: Discuss in your groups
8.2.4. Computing medians
8.2.4.1. Procedure for finding the median
8.2.4.1.1. arrange the numbers in order from least to greatest
8.2.4.1.2. Count the number of elements present in the set
8.2.4.1.2.1.
if n is odd, the median is the middle number
8.2.4.1.2.2.
if n is even, the median is the average of the two middle numbers
8.2.5. Finding Modes
8.2.5.1. number that appears most frequently in a set
8.2.5.2. can be zero, 1, 2, 3, or more of these
8.2.5.3. Now try this 8-5 p. 529: Discuss in your groups
8.2.5.4. Now try this 8-6 p. 529: Discuss in your groups
8.2.6. Choosing the most appropriate average
8.2.6.1. See examples 8-4, 8-5, 8-6 p. 530
8.2.7. Measures of Dispersion – Measures of Spread
8.2.7.1. Range – highest minus lowest
8.2.7.2. Quartiles
8.2.7.2.1. Q1, Q2, Q3
8.2.7.2.2. Related to percentiles
8.2.7.2.3. IQR: Q3 – Q1 = IQR
8.2.7.3. Now try this 8-7 p. 532: Discuss in your groups
8.2.8. Box plots
8.2.8.1. Often called box and whisker plots
8.2.8.2. uses range and quartiles to display data summary graph
8.2.8.3. see fig. 8-19 and 8-20 p. 533
8.2.9. Outliers
8.2.9.1. Expected range using quartiles
8.2.9.1.1. Q1 – 1.5IQR
8.2.9.1.2. Q3 + 1.5IQR
8.2.9.2. Expected range using mean and standard deviation
8.2.9.2.1. x  2s
8.2.9.2.2. x  2s
8.2.9.3. See example 8-8 p. 534-5
8.2.10. Comparing sets of data
8.2.10.1. Now try this 8-8 p. 537: Discuss in your groups
8.2.11. Variation
8.2.11.1. Mean Absolute Deviation (MAD)
8.2.11.1.1. absolute deviation:
8.2.11.1.1.1.
x1  x  x2  x   xn  x
n
 xi  x
8.2.11.1.1.2.
i 1
8.2.11.1.2.
MAD 
x1  x  x2  x 
n
n
8.2.11.1.3.
MAD 
 xi  x
i 1
n
8.2.11.2. Variance
n
 x
8.2.11.2.1.
i 1
i
 x
2
n
 s2
n
 xi2  nx 2
8.2.11.2.2.
i 1
n
 s2
8.2.11.3. Standard Deviation
n
8.2.11.3.1.
s  s2 
 x
i 1
i
s  s2 
8.2.12. Normal distributions
2
n
n
8.2.11.3.2.
 x
 xi2  nx 2
i 1
n
 xn  x
8.2.12.1. Normal curve – see figure 8-25 p. 543
8.2.12.2. sometimes called bell curve
8.2.12.3. See figure 8-26 p. 544
8.2.12.4. must have sample size of at LEAST 30 in order to assume data has a normal
distribution
8.2.12.5. Percentiles – related to quartiles
8.2.12.5.1. 1st quartile is 25th percentile
8.2.12.5.2. 2nd quartile is 50th percentile
8.2.12.5.3. 3rd quartile is 75th percentile
8.2.12.5.4. 4th quartile is 100th percentile
8.2.12.6. Should only be used on very large data sets – not appropriate for use in the
classroom or even for a whole grade level
8.2.12.7. How to interpret percentile: 82nd percentile means approximately 82% of those
taking the test scored lower than the student and approximately 18% had higher
scores
8.2.13. Ongoing Assessment p. 545
8.2.13.1. Home work: 1ace, 3a, 4, 6, 7ac, 9a, 11, 13a, 14, 16, 17ac, 18ac, 19a, 20a, 22,
23ac, 25, 27a, 28, 29ac, 31ab, 34ab, 35ace, 36a, 37ac, 39, 47