Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester 2007-2008

Chapter 2 (part 3) Statistics for Describing Data Lecture 5 Chapter 2 (part 3) Statistics for Describing Data Main Reference: Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Section 3-4 Measures of position

Key Concept This section introduces measures that can be used to compare values from different data sets, or to compare values within the same data set. The most important of these is the concept of the z score.

Definition z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean

Measures of Position z score Sample Population Round z to 2 decimal places

Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2

Definition Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q1 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

divide ranked scores into four equal parts Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts 25% Q3 Q2 Q1 (minimum) (maximum) (median)

Find lower & upper Quartile To fined Q1, first calculate one-quarter of n and add ½ to obtain ¼ n + ½ . Round this to nearest integer. Example 1 1 1 2 3 3 8 11 14 19 19 20 n = 11,then ¼ n + ½ = ¼ (11)+½ = 3.25 rounded off to 3 Q1 = 2 Q3 = 19 Example 2 2 5 5 6 7 10 15 21 21 23 23 25 n = 12,then ¼ n + ½ = ¼ (12)+½ = 3.5 then the Q1 in position 3 & 4 which is (5+6)/2=5.5 Q2 in position 9 & 10 which is (21+23)/2=22

Percentiles Just as there are three quartiles separating data into four parts, there are 99 percentiles denoted P1, P2, . . . P99, which partition the data into 100 groups. Percentile of value x = • 100 number of values less than x total number of values

Converting from the kth Percentile to the Corresponding Data Value Notation n total number of values in the data set k percentile being used

Example 1 Find the percentile corresponding the weight of 0.8143 & find P10, P25 Solution

Semi-interquartile Range: Some Other Statistics Interquartile Range (or IQR): Q3 - Q1 Semi-interquartile Range: 2 Q3 - Q1 Midquartile: 2 Q3 + Q1 10 - 90 Percentile Range: P90 - P10

Recap In this section we have discussed: z Scores z Scores and unusual values Quartiles Percentiles Other statistics

Exploratory Data Analysis (EDA) Section 3-5 Exploratory Data Analysis (EDA)

Key Concept This section discusses outliers, then introduces a new statistical graph called a boxplot, which is helpful for visualizing the distribution of data.

Important Principles An outlier can have a dramatic effect on the mean. An outlier can have a dramatic effect on the standard deviation. An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

Definitions For a set of data, the 5-number summary consists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value. A boxplot is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

Boxplots

Boxplots – cont.

Boxplots – cont.

Boxplots – cont.

Boxplots - Example

Recap In this section we have looked at: Exploratory Data Analysis Effects of outliers 5-number summary Boxplots

General Examples

Example 1 Fine mean, median, mode, midrange Solution

Example 2 Fine Standard deviation, variance for each of the two sample

Example 3

Example 4 Fine the indicated quartile or percentile a) Q1, b) Q3, c) P80, d) P33 Q1 position = ¼ n + ½ = ¼ (36)+½ = 9.5 (between 9th – 10th) Q1= ( 0.8143+0.815 )/2=0.8147 Q3= ( 0.8207+0.8211 )/2=0.8209

Example 5 Draw the boxplot for the following data set Solution

Flash points

Which measure of center is the only one that can be used with data at the nominal level of measurement? Mean Median Mode

Which of the following measures of center is not affected by outliers? Mean Median Mode

Find the mode (s) for the given sample data. 79, 25, 79, 13, 25, 29, 56, 79 79 48.1 42.5 25

Which is not true about the variance? It is the square of the standard deviation. It is a measure of the spread of data. The units of the variance are different from the units of the original data set. It is not affected by outliers.

Weekly sales for a company are $10,000 with a standard deviation of $450. Sales for the past week were $9050. This is Unusually high. Unusually low. About right.

In a data set with a range of 55. 1 to 102 In a data set with a range of 55.1 to 102.8 and 300 observations, there are 207 data points with values less than 88.6. Find the percentile for 88.6. 32 116.03 69 670

H.W 2 Fine mean, median, mode, midrange, range, standard deviation, variance, P30 Then draw the Boxplot Age of US President