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Chapter 2 – Describing Distributions with Numbers Example : Phyllis had 6 assignment grades in her Stat2331 class: 86 88 92 44 89 90. Idea: We want a few numbers that can summarize a list of observations. Measures of Center I. Mean : Sum of all the observations divided by the number of observations in the list. II. Median calculate median:: The middle number when the observations are put in order. To a) sort observations from smallest to largestb) if n is odd (n = number of observations) median = middle value of the sorted list = n^ 2 +^1 th observation up from the bottom of the list c) ifn is even median = mean of middle two observations
Example : Phyllis grades (cont’d) Her mean grade is: Her median grade is: Issue: Which one to use? Q: Does the mean, 81.5, give a good idea of her “typical” grade? What about the median, 88.5?
Suppose we exclude the outlier 44. We have
M x
With the outlier 44 Without the oulier 44 → resistant measure The mean is pulled by extreme observations or outliers. So it is of center. not a
→ The median is not pulled by the outliers. So it is a resistant measure of center. Q she scored but lost this info. Which would be more useful to him: In figuring Phyllis’ grade, her instructor needs to know total HW points
a) her mean score b) her median score
→ When the distribution is skewed: For “ typical ” value, use But if interested in total , use Issue : How do they relate to each other on various distributions? Symmetric:
Right-skewed:
Left-skewed:
Measures of Spread I. Standard deviation (s): measures the “average” distance between each observation and the mean of the data. To calculates: (a) Calculate • Compute squared distance between each observation and the mean of variance (s^2 ):
- the data.Find an “adjusted” average of these distances.
(b) Take square root of the variance to gets.
Ex : Water bills of two restaurants (cont’d). TGIF’s
x i xi − x ( xi − x )^2
Total Variance (s^2 ) = SD (s) = Similarly, forOutback’s we gets^2 = ______ ands = _____.
Behavior & Properties of Standard Deviation
- s measures the spread about the mean and should be used only when the mean is chosen to measure the center, i.e.,
- s is in the same units as the original observations, e.g.,
- shigher the ≥ 0. The more the data are spread out from the mean, the smaller /s.
- If all numbers in a list are same,s = ___.
- How would outliers affect thes?
- s is / is not a resistant measure. Should not be used to describe a Ex : Phyllis HW grades contd.: 86 88 92 44 89 90
With the outlier 44^ x^ s
Without the outlier 44 Ex : Choose 4 numbers from the list 0, 1, …, 10, repeats allowed, such that (a) they have the smallest possible SD.
(b) they have the largest possible SD.
II. Five-Number Summary : Uses five numbers to divide the whole distribution in four equal parts. One-quarter of all the observations are covered between two consecutive numbers.
We can graphically display the five-number summary by a boxplot. A boxplot consists of:
- A central box that spans the distance between Q 1 and Q 3.
- Horizontal lines marking min., median and max.
- Vertical lines extending from the bottom and top of the box tothe min. and max. observations, respectively.
Boxplots of Home run data:
Notes : (1) Boxplots are best used for side-by-side comparison of more than onedistribution.
(2) Make sure that you include a numerical scale while drawing boxplot. (3) For skewed distributions, use five-number summary and boxplots. (4) Boxplots give an indication of the symmetry or skewness of a distribution. In a symmetric distribution, the first and third quartiles are equally distant from the median.
Example : Consider the box plot at the left, which summarizesthe systolic blood pressures of 39 adult males. Q: Estimate the 5-number summary from the boxplot. Min = Q 1 = Q 2 = Q 3 = Max = Fivedistribution. To get an idea about the spread only, we can use number summary and boxplot give us an idea about Inter-Quartile the whole Range (IQR):
Ex: Home Run data IQR for Ruth = IQR for Maris =
Use TI-83 to Calculate Numerical Summaries We will use the list Ldifferent list in the calculator. 1 for illustration, however, the following methods applied to
Step 1. Clear the data list press STAT Æ 4:ClrList Æ put L 1 by pressing 2nd 1 ( or 2nd STAT, and choose L 1 ) Step 2. Enter the data press STAT Æ 1:Edit Æ enter the observations one-by-one into the list L 1 Step 3. Obtain one-variable statistics press STAT press right arrow key to highlight CALC press ENTER for 1-Var Stats enter the name of the list containing your data press press 2nd 1 for L1ENTER
