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1 Probability and Statistics Chapter 2 Notes I Section 2-1 A Steps to Constructing Frequency Distributions 1 Determine number of (may be given to you) a Should be between and classes 2 Find the Range a The the 3 Find the Class Width 1) Use the TI-84 to sort the data a) STAT Edit enter numbers into L1 b) STAT SortA(L1) will put the numbers into ascending order c) STAT SortD(L1) will put the numbers into descending order a divided by the 4 Find the Lower Limits 1) Always round!! a) Even if class width comes out to a, go up one a Begin with the value in your data set, and then add the class width to that to get the next Lower Limit 5 Find the Upper Limits 1) Repeat as many times as needed to get the required number of classes a The Upper Limit of the first class is than the of the 1) Add the class width to each Upper Limit until you have the necessary number of classes 6 Find the Lower Boundaries a Subtract from each Lower Limit (Do NOT round these!) 7 Find the Upper Boundaries a Add to each Upper Limit 8 Find the Midpoints of each class a The of the (Do NOT round) Could also use the means of the boundaries for this 9 Frequency Distribution a Place a in each class for every that b Add up the tally marks these are your for each class 10 Relative Frequencies a Divide the by the of 11 Cumulative Frequencies to find the a The total number of tallies for each class, plus all those that came before 1) The cumulative frequency of the last class must equal the of the total represented by

2 EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes On the calculator, Press the STAT key, and select Edit Enter the 30 numbers into L1 on the calculator Press STAT, select SortA and designate L1 to be sorted 2 nd and the number 1 puts L1 into the parentheses Press STAT, select Edit to see the new list, in ascending order now Fill in the blank table below with the numbers in order Now we will actually fill in the distribution table FILL THIS IN AS WE GO THROGH THE POWERPOINT!!! Max Value: Min Value: Range: Class Width Lower Upper Lower Upper Midpoint Frequency Relative Cumulative Limit Limit Boundary Boundary Frequency Frequency Minimum Value is the First Lower Limit Fill in the first column going down Do NOT go across!! Go DOWN the column! Add the class width to each lower limit to find the next one First Upper Limit is one less than the 2 nd Lower Limit Add class width to each upper limit to find the next one Once you have your classes identified, check to be sure that the maximum value fits in your table If it doesn t fit, you did something wrong! The most common mistake is to forget to round UP on the class width To find the boundaries, subtract one-half unit from the lower limits, and add one-half unit to the upper limit Notice that the class width is now 8, which is what we would expect it to be Count how many of the 30 values fit into each class These are your class frequencies Divide the class frequencies by 30 (the total number of data points) to find each relative frequency The relative frequencies MUST add up to 1 If not, you did something wrong Check your work The cumulative frequency is that class frequency plus the frequencies of all the classes above that class

3 The last cumulative frequency MUST match the number of data points In this case, the last cumulative frequency has to be 30 B Steps to Constructing a Frequency Histogram 1 Label the with the class boundaries 2 Label the with the number of frequencies 3 Draw a with bars that touch, using the from your frequency distribution C Steps to Constructing a Relative Frequency Histogram 1 Label the horizontal axis with the 2 Label the vertical axis with the 3 Draw a bar graph with bars that touch, using the from your frequency distribution D Steps to Constructing an Ogive 1 Label the horizontal axis with the 2 Label the vertical axis with the 3 Place a dot at each that corresponds to that class s a This chart will always end at the total number of data points II Section 2-2 A Stem and Leaf Plot 1 Use the extreme values as your 2 Go through the data points, placing the leaves beside the appropriate stems 3 If you have too many data points, you can use per stem, with consisting of the first line, and on the second line B Dot Plot 1 Use a, numbered from lowest data value to highest a Place a dot on the line at each 1) This allows you to see visually whether you have a tight grouping of data points, and where it is, if it exists C Pie Chart 1 Used to describe a Multiply the you calculated earlier by 360 (the number of degrees in a circle) to find the number of degrees that each class will consist of 1) The calculated number of degrees corresponds to the in the circle a) Use a protractor to draw your angles D Scatter Plot 1 Used to the between two different elements a Place one element on the, and the other on the 1) Graph them as if one was the of an ordered pair and the other was the 2 The the dots are to being, the the relationship

4 a If the slope is upward, the relationship has a correlation b If the slope is downward, the relationship has a correlation 3 To do a Scatter Plot on the TI-84, follow these steps A Turn STAT Plots on 1) 2 nd y=, Enter 2) Highlight Plot On, Enter B) Go to STAT and Edit 1) Enter x-values into L1, and y-values into L2 C) Press the Window button, and set your x-min and x-max values to match the data in L1 1) Repeat for y-min and y-max values to match L2 D) Press graph to see the scatterplot 4 To get the equation of the line of best fit, go to STAT and Calc, then select LinReg (4) A) The slope and y-intercept will be given to you 5 To graph the line with the scatterplot, manually enter the equation into the y= window and press Graph IVSection 2-4 A Measures of Variation 1 Range the between the and (Maximum minus Minimum) a Easy to compute but only uses from a data set 2 Deviation The between the of a and the of the a In a the deviation of x is x μ (Greek letter mu, pronounced moo ) b In a, the deviation of x is x x (pronounced x bar ) c The sum of the of a set of data will always be 3 Population Measures of Variance a Population Variance -- The of the of the deviations, divided by (the number of in the ) 1) Find the, and then (this makes them all positive, so they don t each other out) a) Add up the deviations, and then by the number of data points b Population Standard Deviation The of the population variance 4 Sample Measures of Variance a Sample Variance The of the of the deviations, divided by (one less than the number of data points in the sample) b Sample Standard Deviation The of the sample variance B Empirical Rule 1 symmetric bell-shaped distributions have the following characteristics: a About of data points will occur within standard deviation of the mean b About of data points will occur within standard deviation of the mean c About of data points will occur within standard deviation of the mean C Chebychev s Theorem 1 This applies to distribution, regardless of its

5 a The portion of data lying with k (k > 1) of the is 1 1 k 2 1) For k = 2, at least 1 ¼ = ¾ or of the data will be within standard deviations of the mean 2) For k = 3, at least 1 1/9 = 8/9 or of the data will be within standard deviations of the mean III Section 2-3 A Measures of Central Tendency 1 Mean The of all data points by the of a This one is the one that we most often think of when we say 1) It s also the one most affected by an, (either or ) 2 Median the (or of ) when the data points are a The point which has as many data values as there are 3 Mode The value that happens the (highest ) B Shapes of Distributions 1 Symmetric Data in the, with distribution on either side 2 Uniform Data is across the 3 Skewed Data Named by the a Skewed right means most of the data values are to the ( ) end of the range b Skewed left means that most of the data values are to the ( ) end of the range V Section 2-5 Measures of Position A Quartiles 1 Q 1, Q 2 and Q 3 divide the data into a Q 2 is the same as the, or the b Q 1 is the of the data Q 2 c Q 3 is the of the data Q 2 d If the list of numbers is entered into Stat-Edit on the TI-84, Stat-Calc-1 will give you these values 2 Box and Whisker Plot a runs from to Q 1 b Box runs from, with a line through it at 1) The distance from Q 1 to Q 3 is called the c runs from Q 3 to d To draw a box-and-whisker plot on the TI-84, follow these steps 1) Enter the data values into L1 in STAT Edit 2) Turn on your Stat Plots (2 nd Y=), and select the plot with the box-and-whisker shown 3) Set your window to match the data a) Xmin should be less than your lowest data point b) Xmax should be more than your highest data point

6 B Percentiles C Z-Scores 4) Press graph The box-and-whisker plot should appear a) Press the Trace button and you can see exactly which values make up the Min, Q 1, Median, Q 3, and the Max 1 Divide the data into There are (P 1, P 2, P 3, P 99) a P 50 = = b P 25 = c P 75 = 2 A 63 rd percentile score means that this person did or 63% of the people who took that test 3 The that we did way back in section one can help us find the percentile 1 Also called the, it represents the of away from the mean (z = value mean standard deviation = x μ σ ) 2 A z-score of less than -2 or greater than 2 is considered to be that a data value is a Remember that of data points should be within 2 standard deviations of the mean (if the data is symmetrically distributed) 3 A z-score of less than -3 or greater than 3 is considered to be a Remember that of data points should be within 3 standard deviations of the mean (if the data is symmetrically distributed)

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