Measures of Center

Notation Chapter 3
Section3-2: Measures of Center Suppose we are making a series of observations, n of them, to be exact. Then we write x1 , x 2 , x 3, K , x n as the values we observe. Thus n is the total number of data points, and x 4 (say) is the value of the fourth data point. 1 Example Measures of Center and Spread Suppose we ask five people how many hours of television they watch in a week, and get the following data: Observation 1 2 3 4 5 Data value 5 7 3 38 7 CENTER
That is,
x1 = 5, x 2 = 7, x3, = 3, x 4 = 38, x5 = 7

SPREAD Mean Median Mode Midrange


What is the “center” of these data, and what is the “spread” about that value? Range Inter-quartile range (IQR) Variance/Standard deviation Mean Sample Mean

Mean Median
Mode x Population Mean “x-bar” the average of a set of observations in a sample If the n observations are then Midrange


µ “mu” the average of a set of observation in the population If the N observations are then n x + x 2 + x3 +K+ x n x= 1 = n 5 N ∑x i =1 n i x + x 2 + x 3 +K+ x N x= 1 = N ∑x i i =1 N 6 1 Example: Sample Mean Mean is the balance point of the distribution x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38, x5 = 7 The mean is x + x 2 + x 3 + x 4 + x5 x= 1 = n 5 + 7 + 3 + 38 + 7 x= = 12 5 5 ∑x i i =1 n Median Median
The median M is the midpoint of the distribution (like the median strip in a road)
It is the number such that half of the observations fall above and half fall below. Example: Median Example 2: Median x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38, x5 = 7
Step 1: order the data: 3 5 mean 7 7 Median = 7 If the data are x1 = 5, x 2 = 7, x 3, = 3, x 4 = 38
38 Step 1: order the data: 3 5 7 Median = 38 5+ 7 =6 2 2 Example outlier Data set A: 64 65 66 68 70 71 73

Data set B: 64 65 66 68 70 71 730 median is 68 mean is 68.1

median is still 68 mean is 162 The mean is very sensitive to outliers, while the median is resistant to outliers. Symmetric distributions with no outliers Mean ≈Median Right-skewed distributions Median=68 Comparing the mean and the median Median Mean

describes the center as an average value, where the actual values of the data points play an important role Sensitive to outliers

locates the middle value as the center, and the order of the data is the key to finding it. Not sensitive to outliers Left-skewed distributions Mean < Median Skewness Mean=72 Median< Mean 3 Which measure of center to use?
Another measure of center: Mode We will therefore use the mean as a measure of center for symmetric distributions with no outliers.
Mode is the most frequent value in the data set
Data: 2, 4, 4, 4, 5, 5, 6, 7, 8, 10, 15
Data: 2, 4, 4, 4, 5, 5, 5, 6, 7, 8, 10, 15 Mode = 4
Otherwise, the median will be a more appropriate measure of the center of our data. Mode = 4, 5
Data: 2, 4, 5, 6, 7, 8, 10, 15 No mode And another measure of center: Midrange Summary: Measures of Center

Midrange is the value midway between the maximum and the minimum values in the data set.

midrange =
max + min 2

The two main numerical measures for the center of a distribution are the mean x and the median M. The mean is the average value, while the median is the middle value. The mean is very sensitive to outliers, while the median is resistant to outliers. The mean is an appropriate measure of center only for symmetric distributions with no outliers. In all other cases, the median should be used to describe the center of the distribution. The mode is the most frequent value. The midrange is the average of the max and the min values. 21 Range Interquartile range (IQR) Variance/Standard deviation 23 24 4 Spread


Measures of Spread Spread: how far from the center the data tend range. If all the data points are identical, there would be no spread at all. Numerically, the spread would be zero. Ex.: 5 5 5 5 5 5 5 5 5 5 5 Center: 5 Spread: 0 Range
Range = max. value – min. value
Example: Data: 2, 4, 4, 4, 5, 5, 6, 7, 8, 10, 15 Inter-quartile range (IQR)
the IQR gives the range covered by the MIDDLE 50% of the data Range= max.- min. = 15-2 = 13 How to find the IQR?
Step 1: arrange the data in increasing order
Step 2: find the median How to find the IQR?
Step 3: Find the median of the lower 50% of the data. This is called the first quartile of the distribution and is denoted by Q1. 5 How to find the IQR?
IQR Step 4: Repeat this again for the top 50% of the data. Find the median of the top 50% of the data. This is called the third quartile of the distribution and is denoted by Q3. IQR
The middle 50% of the data falls between Q1 and Q3, and therefore: IQR = Q3 - Q1 Example
Weights of 10 students: 102, 118, 120, 136, 138, 149, 157, 157, 161, 180 Q3 Q1 Q1 M= 138 + 149 = 143.5 2 Q3 IQR = 157-120 = 37 Q1 Q3 Note
Using the IQR to detect outliers The IQR should be used as a measure of spread of a distribution only when the median is used as a measure of center. Median IQR
The 1.5(IQR) Criterion for Outliers
An observation is considered a suspected outlier if it is  below Q1 - 1.5(IQR) or  above Q3 + 1.5(IQR) 6 The 1.5(IQR) Criterion for Outliers Example 1
Weights of 10 students: 102, 118, 120, 136, 138, 149, 157, 157, 161, 215 Q1 M= 138 + 149 = 143.5 2 Q3 IQR = 157-120 = 37 Q3 + 15 . IQR = 157 + 15 . (37) = 212.5 Anything above 212.5? YES. 215 IS an outlier. Five-number summary Example 2 Outliers! Outlier!
Data: -15, 8, 9, 12, 14, 19, 22, 23, 23, 45, 50
To get a quick summary of both center and spread, we consider these five numbers:  Q1 IQR= 23-9 = 14 M Q3   1.5 (IQR)= 1.5 (14) =21  Anything below Q1-1.5(IQR)=9 – 21 = -12? YES! Anything above Q3+1.5(IQR)=23 +21 = 44? YES! Boxplot  Tukey invented another kind of display to show off the five-number summary. It’s called boxplot. Five-number summary Example 1

John Mininum value Q1 Median Q3 Maximum value Weights of 10 students: 102, 118, 120, 136, 138, 149, 157, 157, 161, 180 Q1 Min. 100 110 120 M= 138 + 149 = 143.5 2 130 140 150 Q3 160 Max. 170 180 7 Example 2 Comparing distributions Outlier
Weights of 10 students: 102, 118, 120, 136, 138, 149, 157, 157, 161, 215 Q1 Min. M= 138 + 149 = 143.5 2 Q3
Max. Boxplots are best used for side-byside comparison of more than one distribution. * 100 110 120 130 140 150 160 170 180 Variance and Standard Deviation
Facts about the standard deviation (s) The standard deviation gives the average (or typical distance) between a data point and the mean, x . Variance: Standard deviation: s2 =
( x1 − x ) 2 + ( x2 − x ) 2 +K+( x n − x ) 2 n −1 s measures the spread about the mean and should be used only when the mean is chosen as the measure of center. That is, when the distribution of the data is roughly symmetric with no outliers. Standard deviation Mean s= ( x1 − x ) 2 + ( x 2 − x ) 2 +K+ ( xn − x ) 2 n−1 46 45 Facts about the Standard Deviation (s)



Standard Deviation Sample s.d.: s s is always zero or greater than 0. s = 0 only when there is no spread, i.e., the data values are identical. s gets larger as the spread increases. s has the same units of measurements as the original observations. Like the mean, s is not resistant. It is very sensitive to outliers. s= Population s.d.: σ ( x1 − x ) 2 + ( x 2 − x ) 2 +K+ ( xn − x ) 2 n −1 σ= 47 ( x1 − x ) 2 + ( x 2 − x ) 2 +K+ ( x N − x ) 2 N 48 8 The 68-95-99.7 rule (Empirical Rule) The Empirical Rule for Data with a Bell-Shaped Distribution Measures of Relative Standing 50 The Empirical Rule Standard Normal Distribution
is a Normal distribution with mean 0 and standard deviation 1.
Notation: N (0,1). 52 How to standardize?
Need to change x to z-scores: z= Ordinary/Unusual Values x−µ σ A z-score measures the number of standard deviations that a data value x is from the mean µ. •When x is larger than the mean, z is positive. Ordinary values: -2 ≤ z-score ≤ 2 Unusual values: z-score < -2, or z-score > 2 •When x is smaller than the mean, z is negative. •When x is equal to the mean, z is zero. 53 54 9 Percentiles Summary are measures of location. There are 99 percentiles denoted P1, P2, . . . P99, which divide a set of data into 100 groups with about 1% of the values in each group.

Percentile of value x = number of values less than x • 100 total number of values


Measures of the center of distributions:  Mean  Median  Mode Measures of spread of distributions:  Range  IQR Using IQR to detect outliers—the 1.5(IQR) rule Boxplots Variance/Standard deviation 10