Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 1 Graded Projects

4.1 What is Average? Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

An average is a number that is representative of a group of data. Most common averages: Mean Median Mode Weighted Average will also be discussed. Definitions 3 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Mean The mean, is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is where represents the sum of all the data and n represents the number of pieces of data. 4 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example-find the mean Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $327 $465 $672 $150 $230 5 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Median The median is the value in the middle of a set of ranked data. Determine the median of $327 $465 $672 $150 $230. Rank the data from smallest to largest. $150 $230 $327 $465 $672 middle value (median) 6 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Median (even data) Determine the median of the following set of data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4. Rank the data: There are 10 pieces of data so the median will lie halfway between the two middle pieces the 7 and 8. The median is (7 + 8)/2 = (median) middle value 7 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Mode The mode is the piece of data that occurs most frequently. Determine the mode of the data set: 3, 4, 4, 6, 7, 8, 9, 11, 12, 15 The mode is 4 since it occurs most often. Two other common types of mode are no mode and bimodal. 8 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example The weights of eight Labrador retrievers rounded to the nearest pound are: 85, 92, 88, 75, 94, 88, 84, 101 Determine the a)mean 9 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

weights: 85, 92, 88, 75, 94, 88, 84, 101 b. Median: rank the data 75, 84, 85, 88, 88, 92, 94, 101 The median is 88. c. Mode-the number that occurs most often. The mode is Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

11 Rounding Rule for Statistical Calculations State your answers with one more decimal place of precision than is found in the raw data. If the calculation gives the same number of decimals as the data, one more is optional. Definitions An outlier in a data set is a value that is much higher or much lower than almost all others. A weighted mean (p. 151) accounts for variations in the relative importance of data values. Each data value is assigned a weight and the weighted mean is weighted mean = sum of (each data value x its weight) sum of all weights Page 147

Normal Value for Grades A4.0 B3.0 C2.0 D1.0 F0 If + only is used the the + value adds a.5, so a B+ is 3.5. If a + & - is used check to see how they break it down (schools vary on this). 12 Course(optional)Letter Grade(optional) Value of Grade Credit HoursValue x Credit Hours Computers & Problem Solving B+3.0 Calculus IIB4.0 Discrete MethodsC+3.0 Religions of the EastA3.0 Russian IIB3.0 Total To calculate a weighted average (used for GPA); use a table similar to the following:

4.3 Measures of Variation Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Measures of Position Measures of position are often used to make comparisons. Two measures of position are percentiles (percent) and quartiles (fourths). Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example: Quartiles The weekly grocery bills for 23 families are as follows. Determine Q 1, Q 2, and Q Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example: Quartiles continued Order(Rank) the data:  Q 2 is the median of the entire data set which is 190.  Q 1 is the median of the numbers from 50 to 172 which is 95.  Q 3 is the median of the numbers from 210 to 330 which is 270. The 5 Number Summary is: 50, 95, 190, 270, Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Measures of dispersion or variation are used to indicate the spread of the data. The range is the difference between the highest and lowest values; it indicates the total spread of the data. Range = highest value – lowest value 17

Example: Range Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries. $24,000$26,500$27,000 $28,500$32,000$34,500 $48,000$56,000 $56,750 Range = $56,750 - $24,000 = $32, Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Standard Deviation The standard deviation measures how much the data differ from the mean. It is symbolized with s when it is calculated for a sample, and with  (Greek letter sigma) when it is calculated for a population. Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299 Step 1) Find the mean. 20 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

Example continued, mean = 514 Step 2) Set up the table 21 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 421,5160 (425) 2 = 180, – 514 = (170) 2 = 28, – 514 = (151) 2 = 22, – 514 = (-215) 2 = 46, – 514 = (-234) 2 = 54, – 514 = (-297) 2 = 88, – 514 = (Data - Mean) 2 Data – MeanData (x) 3084Total

Example continued, mean = 514 s = … The standard deviation is $ Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. Step 3) Put values found into the formula.

23 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. Labrador weights Data Value (x)Value – Mean (Value – Mean) – = ( ) 2 = Total

Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. 24 s = The standard deviation is 7.69 rounded.

Empirical Rule Approximately 68% of all the data lie within one standard deviation of the mean (in both directions). Approximately 95% of all the data lie within two standard deviations of the mean (in both directions). Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions). 25 Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

26 Graph from:

4.2 Shapes of Distributions Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

28 Symmetry or Skewness

Normal Distribution (next week) Histogram 29 Line Graph Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them. Number of Modes

4.4 Statistical Paradoxes Page Can't Type? press F11 Can’t Hear? Check: Speakers, Volume or Re-Enter Seminar Put ? in front of Questions so it is easier to see them.

31 LEARNING GOAL Investigate a few common paradoxes that arise in statistics, such as how it is possible that most people who fail a “90% accurate” polygraph test may actually be telling the truth. Better in Each Case, But Worse Overall It is possible for something to appear better in each of two or more group comparisons but actually be worse overall. This occurs because of the way in which the overall results are divided into unequally sized groups.

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