CHAPTER 4 NUMERICAL METHODS FOR DESCRIBING DATA What trends can be determined from individual data sets?

4.1 Describing the Center of a Data Set What is the center of a data set and how can it be found?

Center and Spread Two of the most critical descriptors of a data set Graphical methods such as those in the last chapter give a general impression of both Numerical methods give precise value that can be compared in detail

The three M’s Mean Median Mode Also known as the average Also called the middle Most Frequent

Mean formula for the sample mean x= each piece of data n= number of pieces of data in the data set x i = I indicates the position of the data from within the original data set Always use more accuracy (more decimals) than any one piece of data has. µ is used for the population mean Greek letters are always used for population values

Median The middle value in a list of ordered values – Median has no symbol but is often abbreviated » Med – Middle number if n is odd – Mean of the two middle numbers when n is even

Compare and Contrast of the Mean and Median Median divides the data into two equal parts 50% of the data is on either side of the median Mean is where the fulcrum would cause the “data scale” to balance if the values had weight It is very sensitive to outliers

Balancing the “data scale” Normal/Bell curve mean median Skewed Left Skewed Right

Dichotomy A situation in which there are only 2 possible responses. i.e. Success or Failure The sample proportion of success (p) » Also called the probability – The population proportion is designated by π

Trimmed Mean Makes the mean less susceptible to outliers Order the data Remove the same number of pieces of data from each end Recalculate the mean % x n = number of pieces to be removed from EACH end A small to moderate trim is 5% to 25%

4.1 Homework Page 110 to 112 2, 5, 9, 12, 13, 14, 15 16

4.2 Describing Variability in a Data Set What is data variability and how is it used to determine standard deviation?

Measures of Variability Range = high – low Deviation from the mean= x i – if positive then x i is larger than the mean if negative then x i is smaller than the mean Sample Variance

Since it is a sample (not all the possible data on a subject) and we know that ∑ (x i - )=0 then knowing all but one x i - the other can be found, so we divide by n-1 (has to do with degrees of freedom concept to be discussed later) Why divide by n-1

Sample Standard Deviation “average distance” the items fall from the mean – A small s or s 2 indicates low variability – A high s or s 2 indicates large variability

Population Variance (knowing all the data) Population Standard Deviation compute to the same accuracy as the population

Interquartile Range IQR IQR = upper quartile (Q3) – lower quartile (Q1) Lower quartile (Q1)—the median of the lower half Upper quartile(Q3)—the median of the upper half IF n is odd, the exact median is excluded from the quartiles Used because it is resistant to outliers There is no special name for the population IQR

Uses of the IQR Standard deviation can be approximated by » SD = IQR/1.35 » If SD > IQR/1.35 it suggests heavier or longer tails than the normal curve

Easy method of calculating the SD to avoid round errors use 4 or 5 decimals past the accuracy of the data

Example 20, 15, 12, 18, 17, 15, 17, 16, 18, 25 Reorder 12, 15, 15, 16, 17, 17, 18, 18 20, 25 range = iqr = Median= 17 Q1= 15Q3= 18

continued Find the standard deviation – By hand by simplified rule ixixi X i -(x i - ) 2 x2x totals

By iqr By calculator Press here for calculator

minitab Given: 154, 142, 137, 133, 122, 126, 135, 135, 108, 120, 127, 134, 122 The Minitab output would be: Descriptive Statistics Variable NMean Median TrMean StDev SE Mean Motion Variable Minimum Maximum Q1 Q3 Motion

4.2 Homework Page (by Hand), 20, 22, 23, 24, 26, 27, 28

4.3 Summarizing a Data Set: Boxplots How can single variable data be summarized in graphical format?

Boxplots Can be used for many types of summarizations Iqr = Q3 – Q1 Outlier = data more than 1.5iqr from the end of the box Extreme=data more than 3iqr from the end of the box 25%

Modified Boxplots Whiskers go to the last piece of data that is not an outlier Outlier (closed circle) Extreme Outlier (open circle)

4.3 Homework Page , 30, 31, 32, 33

4.4 Interpreting Center and Variablity: Chebyshev’s Rule, Empirical Rule, z-scores What determinations can be made about the center of the data set?

Chebychev’s Rule One way of determining the percent of data k deviations from the mean (remember that includes above and below the mean) Use at least terminology Tends to underestimate the percentage Applicable to any data set

Uses approximately for its terminology Since empirical rule refers to normal data sets, the percentages can be divided in half for parts above or below the mean Empirical Rule mean % 13.5% 2.35% 95% 99.7%

Z-Scores Measures the number of standard deviations a particular piece of data is from the mean Often called the standardization formula

Compare and Contrast percent vs percentile Percent Percentile The percent that fall at or below the given value Use the position of the value farthest to the left for repeats

example Find the percent and percentile for each – Sue scored 9 out of 10. There were 10 people in the class. Eight people scored 10, Sue scored 9, and one score 0. Percent 9/10 ∙100= 90% Percentile 2/10∙100= 20 th percentile – The Scores were 0, 5, 7, 7, 8, 8, 8, 9, 9, 10 Percent 9/10∙100=90 % Percentile 7/10∙100=70 th percentile

4.4 Homework Page , 36, 38, 39, 40, 42, 44, 46, 47, 48 1 st interval 5 to 10

4.5 Interpreting the results of Statistical Analysis Read section 4.5 Pages 135 to 137 See next slide for review

Review Pages 138 to , 51, 53, 56, 58, 60, 63, 64