MAT110 CH12
TERMS: The z-Score
If x is a data item in a sample with mean and standard deviation s, then the z-score of x is given by
TERMS: Interpreting Measures of Dispersion
A main use of dispersion is to compare the amounts of spread in two (or more) data sets. A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations.
Q?Identify the variable quantity as discrete or continuous. the number of heads in 50 tossed coins? Discrete Continuous
Discrete ** discrete random variable is a random variable that can take on only certain fixed values. The number of heads in 50 tossed coins is discrete because its only possible values are 0, 1, 2, 3, ..., 49, and 50.
TERM: Empirical Rule
Empirical Rule About 68% of all data values of a normal curve lie within 1 standard deviation of the mean (in both directions), and about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations
TERMS: Statistics
In statistics a population, includes all of the items of interest, and a sample, includes some of the items in the population.
TERMS: The study of statistics can be divided into two main areas. #2 Inferential statistics
Inferential statistics, has to do with drawing inferences or conclusions about populations based on information from samples
TERMS: Qualitative data:
The makes of five different automobiles: Toyota, Ford, Nissan, Chevrolet, Honda Quantitative data can be sorted in mathematical order. The number siblings can appear as 1, 1, 2, 2, 3, 3, 3, 4, 5, 8
Q? Refer to the grouped frequency distribution shown at right. Is it possible to identify, based on the data shown in the table, any specific data items that occurred in this sample? choose: No, it's not possible Yes, it is possible
**The correct answer is No, it is not possible. Since each class contains more than one value, there are multiple possibilities for the value of any data items in that class.
Q? The mean clotting time of blood is 7.45 seconds with a standard deviation of 3.6 seconds. What is the probability that an individual's clotting time will be less than 3 seconds or greater than 12 seconds? Assume a normal distribution.
.211 WORK------------
EXAMPLE: Comparing with z-Scores Two students, who take different history classes, had exams on the same day. Jen's score was 83 while Joy's score was 78. Which student did relatively better, given the class data shown below?
Calculate the z-scores: ***SEE PIC Since Joy's z-score is higher, she was positioned relatively higher within her class than Jen was within her class.
TERMS: Population
In statistics a population, includes all of the items of interest
EXAMPLE: Median for a Distribution Find the median for the distribution. VALUE.....1....2...3...4....5 Freq.........4....3...2...6....8
Solution Position of median = The median is the 12th item, which is a 4.
TERMS: Measures of Dispersion
Sometimes we want to look at a measure of dispersion, or spread, of data. Two of the most common measures of dispersion are the range and the standard deviation.
Formula: Convert Percent into Degrees
Take the percentage and make it a decimal example: (100% = 1 or34% = .34, etc.) Take the decimal and multiply it by 360 The resulting number is the degrees of a circle
TERMS: Mean Number of Siblings
Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mean number of siblings for the ten students.
TERMS: Descriptive statistics
The study of statistics can be divided into two main areas. Descriptive statistics, has to do with collecting, organizing, summarizing, and presenting data (information).
Example: Interpreting Measures Two companies, A and B, sell small packs of sugar for coffee. The mean and standard deviation for samples from each company are given below. Which company consistently provides more sugar in their packs? Which company fills its packs more consistently?
We infer that Company A most likely provides more sugar than Company B (greater mean). We also infer that Company B is more consistent than Company A (smaller standard deviation).
TERMS: Sample
a sample, includes some of the items in the population.
Q? A doctor randomly selects 40 of his patients and obtains the following data regarding their serum HDL cholesterol. 34, 51, 48, 37, 41, 63, 65, 42, 53, 58, 46, 41, 66, 36, 44, 53, 52, 63, 51, 63, 42, 54, 36, 46, 41, 63, 54, 52, 43,
(a) Compute the mean and the median serum HDL. Mean=49.6 Median=48 What type of dist. histogram?
Q? Deciles- On the right are ratings for passers in the National and American conferences of a football league. Find the 7th decile for both conferences. ...........................Rating points Martinez........99.5 Rutz.................99.4 Westin.............96.7 Abdullah.......95.3 Lawrence......95.2 Perez..............89.3 Schmidt.........89.2 Emmanuel....86.2 Jarvis..............82.1 Newman......81.6 Hoehne........80.8 Burns.............80.7 Hardy............79.5 Morgan........78.3 Jefferson......78.1 Kazinsky.......77.4 Roddick.......77.2 Singh.............76.8 Goldman.....76.7 Harvey..........75.3
14 WORK---------------------------- The first task is to rank the ratings from both conferences from highest to lowest. What is the highest rating? 99.5 second highest rating is 99.4 the third is 96.7 What is the 4 th rating on your ordered list? 95.3 The combined ratings, in order from highest to lowest, are listed below. 99.5, 99.4, 96.7, 95.3, 95.2, 89.3, 89.2, 86.2, 82.1, 81.6, 80.8, 80.7, 79.5, 78.3, 78.1, 77.4, 77.2, 76.8, 76.7, 75.3 The 7 thdecile is the number below which 70% of the ratings lie. What is 70 % of 20 ratings? 14 **In your ranked list, 14 scores must lie below the 7 th decile. Therefore, the 7 th decile is the 15 th number, counting from the bottom of the list. What is the 7th decile? 89.3
Q? MidRange - for data below... formulat = Highest + lowest / 2 Passer.................Rating Points Martinez................100.6 Washington..........97.2 Abdoullah.............94.2 Schmidt...................85.6 Jarvis........................83.3 Jefferson.................81.1 Roddick..................78.9 Singh......................77.6 Goldman..............76.4 Harvey...................75.4 The midrange is found using the following formula. **Find the minimum and maximum items and substitute these values into the formula.
88
Q? Assume that a normal distribution of data has a mean of 18 and a standard deviation of 3. Use the empirical rule to find the percentage of values that lie above 12. First, compare the score (12) to the mean (18). Find the absolute value of the difference between the scores.
97.5% WORK------------------ 12-18 = 6 If the standard deviation is 3, calculate how many standard deviations are represented by the difference 6 6/3 (fraction) = 2 According to the empirical rule, 95% of the values in this distribution will lie within 2 standard deviation(s) of the mean. (see pic) The task, however, is to find the percentages that lie above 12. Note on the diagram the percentage of values that lie above and below 2 standard deviations from the mean. Be sure to add 95 % to that percentage. What percentage lie above 12 ? 97.5% (2.5% + 95%)
Q? A company has tested a new cellular battery. The mean number of hours that a newly charged battery remains charged is 42 hours, with a standard deviation of 4 hours. What is the percent of batteries that will remain charged more than 34 hours?
97.7% WORK---------------- 42-34=4 find the z-score. Check the table to find the area under the normal curve for that z-score. Convert to a percent. Since 50% of the batteries lie on each side of the mean, add 50 % to that percentage for your final answer. 50%+47.7% = 97.7 (chart 2 = .477 )
TERMS: Range
For any set of data, the range of the set is given by Range = (greatest value in set) - (least value in set).
TERMS: Chebyshev's Theorem
For any set of numbers, regardless of how they are distributed, the fraction of them that lie within k standard deviations of their mean (where k > 1) is at least ***formula in pic
TERMS: Circle Graphs A graphical alternative to the bar graph is the circle graph, or pie chart, which uses a circle to represent all the categories and divides the circle into sectors, or wedges (like pieces of pie), whose sizes show the relative magnitude of the categories. The angle around the entire circle measures 360°. **For example, a category representing 20% of the whole should correspond to a sector whose central angle is 20% of 360° which is 72°.
Formula: % /100=?? *360 = answer in degrees
TERMS: Measures of Position
In some cases we are interested in certain individual items in the data set, rather than in the set as a whole. We need a way of measuring how an item fits into the collection, how it compares to other items in the collection, or even how it compares to another item in another collection. There are several common ways of creating such measures and they are usually called measures of position.
TERMS: Inferential statistics
Inferential statistics, has to do with drawing inferences or conclusions about populations based on information from samples
TERMS: Basic Concepts
Information that has been collected but not yet organized or processed is called raw data. It is often quantitative (or numerical), but can also be qualitative (or nonnumerical).
TERMS: Position of the Median in a Frequency Distribution
Notice that this formula gives the position, and not the actual value.
Q? On the right are ratings for passers in a football league. Find the three quartiles for the ratings. Passer...............Rating Points Martinez...................100.5 Washington..............96.2 Abdoullah.................93.1 Schmidt.......................88.1 Jarvis.............................83.5 Jefferson......................81.5 Roddick.......................78.3 Singh.............................77.7 Goldman.....................76.1 Harvey..........................75.3
Q 1 = 77.7 Q 2 = 82.5 Q 3 = 93.1 (took off 2 highest & lowest scores)
Q? In a calculus class, Jack Hartig scored 9 on a quiz for which the class mean and standard deviation were 7.7 and 2.2, respectively. Norm Alpina scored 6 on another quiz for which the class mean and standard deviation were 4.4 and 2.3, respectively. Relatively speaking, which student did better? Make use of z-scores. Relatively speaking, ??scored better on his quiz. (Jack or Norm)
Relatively speaking, ?? NORM scored better on his quiz. WORK--------------- Calculate the z-score of each student. The student with the higher z-score did relatively better on his quiz when compared with his classmate.
Q? In a calculus class, Jack Hartig scored 8 on a quiz for which the class mean and standard deviation were 6.9 and 1.9, respectively. Norm Alpina scored 6 on another quiz for which the class mean and standard deviation were 4.2 and 2.2, respectively. Relatively speaking, ??scored better on his quiz. (Jack or Norm)
Relatively speaking, Norm scored better on his quiz.
EXAMPLE: Mode for a Set Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mode for the number of siblings.
Solution 3, 2, 2, 1, 3, 6, 3, 3, 4, 2 The mode for the number of siblings is 3.
EXAMPLE: Median Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 1, 6, 3, 3, 4, 2. Find the median number of siblings for the ten students.
Solution In order: 1, 1, 2, 2, 2, 3, 3, 3, 4, 6 Median = (2+3)/2 = 2.5
EXAMPLE: Mode for a Distribution Find the median for the distribution. VALUE.....1....2...3...4....5 Freq.........4....3...2...6....8
Solution The mode is 5 since it has the highest frequency (8).
Example: Range of Sets The two sets below have the same mean and median (7). Find the range of each set. Set A.........1....2....7....12....13 Set B.........5....6....7....8.....9
Solution: Range of Set A: 13 - 1 = 12. Range of Set B: 9 - 5 = 4.
EXAMPLE: Deviations from the Mean Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13.
Solution: The mean is 7. Subtract to find deviation. Data Value.......1....2....8....11....13 Deviation........-6....-5....1....4....6 The sum of the deviations for a set is always 0.
Q? Approaching midseason, the teams in the National Basketball Association had won the following numbers of games. Construct a stem-and-leaf display for the given data. In each case, treat the ones digits as the leaves. For any single-digit data, use a stem of 0. 22, 34, 23, 42, 43, 24, 47, 27, 29, 51, 62, 53, 55, 63, 59, 64, 27, 62, 34, 59
Stem-and-Leaf Displays
Q? To get a B in math, Alexandria Pappas must average 80 on five tests. Scores on the first four tests were 83, 77, 84, and 76. What is the lowest score that she can get on the last test and still get a B?
The lowest possible score for getting a B is 80
TERMS: Quantitative data:
The number of siblings in ten different families: 3, 1, 2, 1, 5, 4, 3, 3, 8, 2
TERMS: Median Another measure of central tendency, which is not so sensitive to extreme values, is the median. This measure divides a group of numbers into two parts, with half the numbers below the median and half above it.
To find the median of a group of items: Step 1 Rank the items. Step2 If the number of items is odd, the median is the middle item in the list. Step 3 If the number of items is even, the median is the mean of the two middle numbers.
Q? Suppose that the life expectancy of a certain brand of nondefective light bulbs is normally distributed, with a mean life of 1200 hr and a standard deviation of 150 hr. If 80,000 of these bulbs are produced, how many can be expected to last at least 1200 hr?
Z =40,000 WORK-------------- Step 1: The area under the normal curve is 1. The z-table shows how much area is under the curve between the mean and each z-score. 1200-1200/150 = 0 Step 2: The area under the normal curve is equal to 1, and the normal curve is symmetric about the vertical line through the mean (where z=0). Therefore, P(z> 0)= .5 (Divided the 1 in half) Step 3: The number of light bulbs that would be expected to last at least 1200 hours is 80,000 *0.5 = 40000 light bulbs.
Q? Assume that a normal distribution of data has a mean of 12 and a standard deviation of 3. Use the empirical rule to find the percentage of values that lie above 18. What percentage of values lie above 18?
2.5
EXAMPLE: Box Plot Construct a box plot for the weekly study times data shown below. 1.................5.....8 2.................0.....7...8...9...9 3.................2.....6...6...7 4.................0.....2...2...7...9 5.................1.....5...6 6.................6
The minimum and maximum items are 15 and 66. ***SEE PIC
TERMS: Symmetry in Data Sets
The most useful way to analyze a data set often depends on whether the distribution is symmetric or non-symmetric. In a "symmetric" distribution, as we move out from a central point, the pattern of frequencies is the same (or nearly so) to the left and right. In a "non-symmetric" distribution, the patterns to the left and right are different.
TERMS: Weighted Mean
The weighted mean of n numbers x1, x2,..., xn, that are weighted by the respective factors f1, f2,..., fn is given by the formula
Q? On the right are the numbers of customers served by a restaurant on 40 consecutive days. (The numbers have been ranked lowest to highest.) Find the 85th percentile is?? 46, 48, 48, 49, 50, 52, 53, 56, 56, 58, 61, 64, 66, 68, 70, 70, 70, 71, 73, 74, 76, 77, 81, 85, 87, 87, 88, 88, 88
the 85 th percentile? 86 WORK-------------------- The 85th percentile of items is the number below which 85% of the items lie. Since there are 40 items on the list, first find 85% of 40 0.85*40 = 34 There will be 34 items below the 85 th percentile. The next item (the 35 th item in # list 34+1) will represent the 85 th percentile. 34+1 What is the 85 th percentile? 86
Q? Find the percent of the total area under the standard normal curve between the following z-scores. z = - 1.6 and z = - 0.65 percent of the total area between z = -1.6 and z = -0.65 %.
20
TERMS: Visual Displays of Data
Basic Concepts Frequency Distributions Grouped Frequency Distributions Stem-and-Leaf Displays Bar Graphs, Circle Graphs, and Line Graphs
TERMS: Coefficient of Variation
The coefficient of variation is not strictly a measure of dispersion, it combines central tendency and dispersion. It expresses the standard deviation as a percentage of the mean.
TERMS: Non-symmetric Distributions
A non-symmetric distribution with a tail extending out to the left, shaped like a J, is called skewed to the left. If the tail extends out to the right, the distribution is skewed to the right.
EXAMPLE: Empirical Rule Suppose that 280 sociology students take an exam and that the distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean.
A total of 95% of all scores lie within 2 standard deviations of the mean. (.95)(280)= 266 scores
Q? Explain whether the interquartile range measures central tendency, dispersion, or position, and why. Interquartile range = Q 3 - Q 1 A. The interquartile range measures dispersion, because it characterizes how far the dataset is spread out from a center point. B. The interquartile range measures position, because it describes the relative location of a particular item within the data set. C. The interquartile range measures central tendency, because it is a central value representative of an entire set of numbers.
A. The interquartile range measures dispersion, because it characterizes how far the dataset is spread out from a center point. ***The interquartile range measures dispersion because it indicates the size of the span in which half of the data is located.
Q? Find the grade point average. Assume that the grade point values are 4.0 for an A, 3.0 for a B, and so on. Course.................Credits...............Grade Math........................5............................A English...................4............................B Physics...................4............................B German.................2............................C
AVG GPA 3.2
TERM: Explain whether the Midrange measures central tendency, dispersion, or position, and why. ......................Minimum item + maximum item midrange= ----------------------------------------- .............................................. 2 Choose the correct answer below. A. The midrange measures position, because it describes the relative location of a particular item within the data set. B. The midrange measures central tendency, because it is a central value representative of an entire set of numbers. C. The midrange measures dispersion, because it indicates how much the dataset is spread out from a center point.
B. The midrange measures central tendency, because it is a central value representative of an entire set of numbers. **The midrange is the mean of the greatest and least numbers in a data set. It is a central point that to some degree represents the entire data set.
TERMS: Standard Deviation The variance is found by summing the squares of the deviations and dividing that sum by n - 1 (since it is a sample instead of a population). The square root of the variance gives a kind of average of the deviations from the mean, which is called a sample standard deviation. It is denoted by the letter s. (The standard deviation of a population is denoted the lowercase Greek letter sigma.)
Calculation of Standard Deviation Let a sample of n numbers x1, x2,...xn have mean Then the sample standard deviation, s, of the numbers is given by The individual steps involved in this calculation are as follows 1 Calculate the mean of the numbers. 2 Find the deviations from the mean. 3 Square each deviation. 4 Sum the squared deviations. 5 Divide the sum in Step 4 by n - 1. 6 Take the square root of the quotient in Step 5.
Q? Identify the variable quantity as discrete or continuous. the average weight of babies born in a week? Discrete Continuous
Continuous **Examples: A person's height: could be any value (within the range of human heights), not just certain fixed heights, Time in a race: you could even measure it to fractions of a second, A dog's weight, The length of a leaf,
TERMS: Deciles and Quartiles
Deciles are the nine values (denoted D1, D2,..., D9) along the scale that divide a data set into ten (approximately) equal parts, and quartiles are the three values (Q1, Q2, Q3) that divide the data set into four (approximately) equal parts.
TERMS: The study of statistics can be divided into two main areas. #1 Descriptive statistics
Descriptive statistics, has to do with collecting, organizing, summarizing, and presenting data (information).
TERMS: Measures of Central Tendency
For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers in the set tend to cluster, a kind of "middle" number or a measure of central tendency. Three such measures are discussed in this section.
TERMS: Mean
The mean (more properly called the arithmetic mean) of a set of data items is found by adding up all the items and then dividing the sum by the number of items. (The mean is what most people associate with the word "average.") The mean of a sample is denoted (read "x bar"), while the mean of a complete population is denoted (the lower case Greek letter mu).
TERMS: Mode
The mode of a data set is the value that occurs the most often. Sometimes, a distribution is bimodal (literally, "two modes"). In a large distribution, this term is commonly applied even when the two modes do not have exactly the same frequency
TERMS: Central Tendency from Stem-and-Leaf Displays
We can calculate measures of central tendency from a stem-and-leaf display. The median and mode are easily identified when the "leaves" are ranked (in numerical order) on their "stems." !! ***SEE PIC !!
Example: Chebyshev's Theorem What is the minimum percentage of the items in a data set which lie within 3 standard deviations of the mean?
With k = 3, we calculate *** formula in pic
Q? If your calculator finds both kinds of standard deviation, the sample standard deviation and the population standard deviation, which of the two will be a larger number for a given set of data? (Hint: Recall the difference between how the two standard deviations are calculated.) Fill in the blank below.
The "sample standard deviation" will be a larger number for a given set of data.
Q? Quartiles- Consider a sample of ages of 100 executives.
(a) Order the data and find the 1, 2, 3 quartiles. Q 1= 43 Q 2=50 Q 3=57 (b) Draw a box-and-whisker plot that represents the data set. Choose the correct box-and-whisker plot below. A,B,C,D (look at pic) (c) Interpret the results in the context of the data. Choose the correct answer below. A. Half of the ages are between 43 and 57 years. (correct) B. Half of the ages are between 50 and 57 years. C. Half of the ages are between 43 and 50 years. (d) On the basis of this sample, at what age would you expect to be an executive? You would expect to be an executive at ?? = 50 years. (e) Which age groups, if any, can be considered unusual? Choose the correct group below. A. 20-29, 70-79 and 80-89 (correct.) B. 40-49 and 80-89 C. 50-59, 60-69 and 80-89 D. There is no group that is unusual.
Q? The table on the right shows last initials of basketball players and the number of games played by each. Find the z-score for player C's games played. Player........Games Played A..................73 B.................. 79 C..................71 D..................78 E.................. 76 F.................. 78 G..................71 H..................81 J.................. 73 K.................. 77
-1.34 WORK----------- Find the z-score of a data item x in a distribution of items with the mean x and standard deviation s by using the formula What is the mean x of the games played? 75.7 (Round to the nearest tenth as needed.) Calculate the deviations from the mean and square each deviation.Check your calculation of (77 -75.7 )2 First, subtract 75.7- 77 Then square the result. 1.69 Now find the sum of the squares of the deviations. 7.29 +10.89 +22.09 +5.29 +0.09+5.29 +22.09 +28.09 +7.29+ 1.69 = 110.1 Divide the sum by 9, since there are 10 total items in the list. Take the square root of the quotient. 110.1/9 sq root = 3.50 Now use the mean (x = 75.7) and the standard deviation (s=3.50) to find the z-score for player C(x = 71). = -1.34
Q? On the right are the numbers of customers served by a restaurant on 40 consecutive days. (The numbers have been ranked lowest to highest.) Find the 15 th percentile The number of customers representing the 15 th percentile is??? 46, 48, 48, 49, 53, 56, 57, 61, 64, 66, 66, 68, 69, 70, 70, 71, 74, 75, 76, 76, 79, 81, 83, 87, 87, 88, 88
15 th percentile is? 54 WORK--------- .15*40= 6 6+1= 7th in list
Q? Find the percent of area under a normal curve between the mean and the given number of standard deviations from the mean. (Note that positive indicates above the mean, while negative indicates below the mean.) negative 1.20 First, draw a sketch showing the desired area. see pic The desired area will be between z =0 and z= - 1.20
38.5% WORK--------------The desired area will be between z=0 and z=- 1.20. Use a table of normal distribution areas. This area is 0.385 This is the fraction of the total area under the curve. As a percentage, 0.385 is 38.5%
Example: Percentiles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the 40th percentile
The 40th percentile can be taken as the item below which 40 percent of the items are ranked. Since 40 percent of 30 (count of data) is (.40)(30) = 12, we take the thirteenth item, or 75, as the fortieth percentile.
Q? Is it possible to compute the actual standard deviation for this sample? Class Limits ....................Frequency f 21-25.........................................5 26-30.........................................8 31-35.........................................3 36-40.........................................21 41-45.........................................12 46-50.........................................35 51-55.........................................38 56-60.........................................20 A. The standard deviation can be calculated for the given sample. B. The standard deviation cannot be calculated for the given sample. C. It is impossible to tell.
B. The standard deviation cannot be calculated for the given sample.
Q? A survey asked American workers how they were trained for their jobs. The percentages who responded in various categories are shown in the table below. Use the information in the table to draw a circle graph. -------------------------------------------- Label.....Principal Source Training.........% -------------------------------------------- A............Trained in school..........................19% B............Informal on job ............................34% C............Formal T from Emp......................17% D............Trained Military...............................28% E............No Particular training.................. 2%
Circle Graph
TERMS: The Box Plot A box plot, or box-and-whisker plot, involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and third quartiles (measures of position), all incorporated into a simple visual display.
For a given set of data, a box plot (or box-and-whisker plot) consists of a rectangular box positioned above a numerical scale, extending from Q1 to Q3, with the value of Q2 (the median) indicated within the box, and with "whiskers" (line segments) extending to the left and right from the box out to the minimum and maximum data items.
TERMS: Finding Quartiles
For any set of data (ranked in order from least to greatest): The second quartile, Q2, is just the median. The first quartile, Q1, is the median of all items below Q2. The third quartile, Q3, is the median of all items above Q2.
Q? Frequency Distributions Example The ten students in a math class were polled as to the number of siblings in their individual families. Construct a frequency distribution and a relative frequency distribution for the responses below. 3, 2, 2, 1, 3, 4, 3, 3, 4, 2
Number x --- Freq f ----Relative Freq f/n (%) 1 ----------------- 1 ------------- 1/10 = 10% 2 -----------------3 ------------- 3/10 = 30% 3 -----------------4 ------------- 4/10 = 40% 4 -----------------2 ------------- 2/10 = 20%
TERMS: Frequency Distributions When a data set includes many repeated items, it can be organized into a frequency distribution, which lists the distinct values (x) along with their frequencies (f ). It is also helpful to show the relative frequency of each distinct item. This is the fraction, or percentage, of the data set represented by each item.
Number x Frequency f Relative Freq f/n (%) 1 ----------------- 1 ------------- 1/10 = 10% 2 -----------------3 ------------- 3/10 = 30% 3 -----------------4 ------------- 4/10 = 40% 4 -----------------2 ------------- 2/10 = 20% work--------------------- add all frequency # = n
TERMS: Standard deviation from the mean
One of the most useful measures of dispersion, the standard deviation, is based on deviations from the mean of the data.
EXAMPLE: Standard Deviation Find the standard deviation of the sample 1, 2, 8, 11, 13 Data Value...........1........2......8.....11......13 Deviation............-6.....-5......1.......4......6 (Deviation)2.......36....25.....1......16....36
The mean is 7 Sum = 36 + 25 + 1 + 16 + 36 = 114 Divide by n - 1 with n = 5 (***SEE PIC) Take the square root:
Q? The figure at right contains the salaries of the employees at a clothing store. What is the mean salary?
The mean salary is $17960 **To find the mean salary, you must multiply each salary times the number of employees earning that salary. Then divide the sum of those products by the total number of employees.
Example: Deciles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the sixth decile
The sixth decile is the 60th percentile. Since 60 percent of 30 is (.60)(30) = 18, we take the nineteenth item, or 82, as the sixth decile.
TERMS: Stem and Leaf Displays
The tens digits to the left of the vertical line, are the "stems," while the corresponding ones digits are the "leaves." The stem and leaf conveys the impressions that a histogram would without a drawing. It also preserves the exact data values.
Example: Quartiles The following are test scores (out of 100) for a particular math class. 44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100 Find the three quartiles
The two middle numbers are 78 and 79 so Q2 = (78 + 79)/2 = 78.5. There are 15 numbers above and 15 numbers below Q2, the middle number for the lower group is Q1 = 72, and for the upper group is Q3 = 88
Example: Comparing Samples Compare the dispersions in the two samples A and B. A: 12, 13, 16, 18, 18, 20 B: 125, 131, 144, 158, 168, 193
The values on the next slide are computed using a calculator and the formula for coefficient of variation. ***SEE PIC Sample B has a larger dispersion than sample A, but sample A has the larger relative dispersion (coefficient of variation).
Q? Circle Graphs Consider the chart to the right. What is the greatest single expense category? To the nearest degree, what is the central angle of that category's sector? Which is the largest expense category? Choose the correct answer below. ---------------- A. Net Interest is the single largest expense category. B. Social Security is the single largest expense category. C. Defense is the single largest expense category. D. Medicare & Medicaid is the single largest expense category.
WORK---------------- Step: The measure of a circle is 360 degrees. Each sector, or wedge, of a circle graph represents a category. To find the measure of a sector's central angle, find the percent of 360º which corresponds to that category's percent. D. Medicare & Medicaid is the single largest expense category. ----------------------------- The Medicare & Medicaid category represents 25 % of Federal spending. ---write the percent as a decimal. 25/100=?? .25 ---Multiply. 0.25*360 degrees = ?? 90
Q? The table on the right shows last initials of basketball players and the number of games played by each. Find the z-score for player D's games played. Player Games Played A...........................72 B........................... 79 C........................... 74 D........................... 78 E........................... 75 F........................... 80 G........................... 71 H........................... 81 J........................... 73 K........................... 79
What is player D's z-score? .50 WORK---------------------- step 1: Find the z-score of a data item x in a distribution of items with the mean x and standard deviation s by using the formula z = x-x(overbar) / s Step 2: What is the mean(avg) x of the games played? 76.2 Calculate the deviations from the mean and square each deviation. Calculate the deviations from the mean and square each deviation. x (x -x) overbar 2(sq) 72.......17.64 (formula= (72*7.62)2 squared 79.......7.84 74.......4.84 78.......3.24 75.......1.44 80.......14.44 71.......27.04 81.......23.04 73.......10.24 Step 3: Now find the sum of the squares of the deviations. 17.64 +7.84 +4.84 +3.24 ++14.44 +27.04 +23.04 +10.24+7.84= 117.6 Divide the sum by 9, since there are 10 total items in the list. 117.6/9 (sqrt) = 3.61 Step 4: Now use the mean (x = 76.2) and the standard deviation (=3.61) to find the z-score for player D (x = 78). z=(x -x) overbar/s = 78-76.2/3.61 = .50
TERMS: Percentiles
When you take a standardized test taken by larger numbers of students, your raw score is usually converted to a percentile score. If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn. % given * count of data = answer +1
Q? Find the z-score that best satisfies the condition. 23 % of the total area is to the left of z. Sketch a graph showing 23 % of the area to the left of z. First, should z be positive or negative? Negative Positive The graph should look as shown. (see pic) There should be 27 % of the area between z=0 and the desired z-score. This corresponds to an area of 0.27 The area between z=0 and z=-0.74 has an area of 0.27
z= -.74 has an area of 0.27 WORK-------------- First, should z be positive or negative? Negative
Q? Two brands of car batteries, both carrying 6-year warranties, were sampled and tested under controlled conditions. Five of each brand failed after the number of months shown. Calculate a) both sample means and b) both sample standard deviations. Decide c) which brand battery lasts longer and d) which brand has the more consistent lifetime. Brand A...................Brand B 75...............................68 68...............................65 68...............................66 75...............................67 72...............................60
a) What are the sample means? x A= 71.6 x B= 65.2 b) What are the sample standard deviations? A=3.51 B=3.11 c) Which brand battery apparently lasts longer A or B? Brand A d) Which brand battery has the more consistent lifetime A or B? Brand B
