Graphs

Types of Graphs  Bar Graphs  Pie Charts  Dotplots  Stem and Leaf Plots  Histograms  Box Plots  Scatter Plots  Normal Curves

Requirements for ALL Graphs 4 Title 4 Label for each axis 4 Scale WITH units –Consistent scale within entire range of data 4 Legend if comparative graph 4 As large as possible (use ALL provided space)

Bar Graphs and Pie Charts 4 Can be used for categorical data 4 Bars SHOULD NOT touch 4 Used frequently 4 Must know how to read and interpret but knowing how to draw is not a requirement for AP Statistics

Bar Graphs ItemAmount Tuition fees$5,000 Room and board 9,000 Books and lab2,000 Clothes/cleanin g 1,000 Transportation2,000 Insurance and miscellaneous 1,000

Summarizing Data w/ Bar graph of Categorical Data

Pie Charts ItemAmount Tuition fees$5,000 Room and board9,000 Books and lab2,000 Clothes/cleaning1,000 Transportation2,000 Insurance and miscellaneous 1,000

Dot Plots 4 The most commonly used graph in 1 st level Statistics classes 4 Used to plot frequency of discrete data. 4 Dots must be lined up horizontally 4 Vertical axis is frequency of occurrence 4 Advantages –integrity of data maintained –Easy to see distribution of data

Dotplots ItemAmount Tuition fees$5,000 Room and board9,000 Books and lab2,000 Clothes/cleaning1,000 Transportation2,000 Insurance and miscellaneous 1,000

Stem and Leaf Plots Stem is a place holder - in this case, 10’s Leaf is the last digit Test Scores out of 105 StemLeaf Legend 8 | 1 = test score of 81

Comparative Stem and Leaf Plot Price of Homes in Ten Thousands Legend 18 | 1 = home price of $181,000 West El Paso Central El Paso

Stem and Leaf Plots 4 Integrity of data maintained 4 Easy to visualize distribution of data 4 Comparative plots work well

Box Plots 4 Data is organized from smallest to largest. 4 Scale is based on data 4 Width of each segment of box indicates density of data –Wide segment means data frequency is not as dense 4 25% of the data is in each quartile 4 Quartiles can be found by using 1VARSTAT on calculator

How to construct 4 find five-number summary Min Q1 Med Q3 Max 4 draw box from Q1 to Q3 4 draw median in the box 4 extend whiskers to min & max

Modified Box Plots Verbal SAT scores 280, 340, 440, 490, 520, 540, 560, 560, 580, 580, 600, 610, 630, 650, 660, 680, 710, 730, 740, 740 Q1 530 Median 590 Q3 670 Outlier Min 280 Max 740

Modified boxplots 4 display outliers 4 fences mark off outliers 4 whiskers extend to largest (smallest) data value inside the fence ALWAYS use modified boxplots in this class!!!

Calculating Outliers 4 IQR = Q3 – Q1 (IQR is a number, not a range of numbers) –IQR is the range of the middle 50% of the data 4 Outliers occur outside of “fences” –Data smaller than Q1 – 1.5*(IQR) –Data larger than Q *(IQR) –Each outlier is indicated with a dot, or asterix

Outlier Fence Q1 – 1.5(IQR)Q (IQR) Any observation outside this fence is an outlier! Put a dot for the outliers. Interquartile Range (IQR) – is the range (length) of the box Q3 - Q1

Modified Boxplot... Draw the “whisker” from the quartiles to the observation that is within the fence!

Boxplots 4 Organizes data into segments which is easy to quickly analyze 4 Useful for comparative displays 4 Convenient display of outliers 4 Great for large data sets –Should not be used for data sets smaller than 10 4 Raw data is lost –Can only estimate mean

Histograms 4 They are NOT bar graphs 4 Used for discrete AND continuous data 4 A frequency chart must be created –Intervals defined Example of interval for continuous 0 ≤ x < 10, 10 ≤ x < 20 Example of interval for discrete 0 ≤ x ≤ 9, 10 ≤ x ≤ 19 –Count of data that falls into each interval is recorded 4 The vertical axis is Frequency 4 Horizontal axis is based off of intervals

4 Example Frequency Chart IntervalFreq 0 ≤ x < 58 5 ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < 352

Histograms ClassInterval Freque ncy 1$1-$58 2$6-$106 3$11-$154 4$16-$202 5$21-$254 6$26-$306 7$31-$352

Histograms 4 Easy to visualize distribution of data 4 Raw Data is lost 4 Can estimate mean, median, standard deviation –Use calculator – input in list one the median value of each bar –In list two, input each frequency (height of bar) –1VARSTAT using L2 as FREQ LIST

O-Give Curves (S – Curves) 4 Created from Cumulative Frequency Charts IntervalFreq Cum Freq Rel Cum Freq 0 ≤ x < 5444/60=6.7% 5 ≤ x < /60=22.7% 10 ≤ x < /60=46.7% 15 ≤ x < % 20 ≤ x < % 25 ≤ x < % 30 ≤ x < %

Cumulative Frequency Plot

O-Give Curves 4 Useful for estimating percentile values 4 For example, using the previous page, the 40% percentile value is: –On vertical axis find.4*60=24 –Trace over horizontally along 24 to find intersection on graph –Go down and find value on x-axis ≈ 14

Scatter Plots 4 Different than Dot Plots 4 Made with Bi-Variate Data –Two pieces of information for one “object” 4 Least Squares Regression Lines –Will be covered in a section by itself

Scatter Plots Candidate Days studied Score earned

Normal Curve 4 Called a Density Curve –The height signifies more frequency 4 Used with Continuous data 4 Can be used with Discrete Data also –Just imagine a dot plot or histogram where the tops of each bar or column are connected.

Density Curves Can be created by smoothing histograms ALWAYS on or above the horizontal axis Has an area of exactly one underneath it Describes the proportion of observations that fall within a range of values Is often a description of the overall distribution Uses  &  to represent the mean & standard deviation

Normal Curve Bell-shaped, symmetrical curve Transition points between cupping upward & downward occur at  +  and  –  As the standard deviation increases, the curve flattens & spreads As the standard deviation decreases, the curve gets taller & thinner

Empirical Rule Approximately 68% of the observations are within 1  of  Approximately 95% of the observations are within 2  of  Approximately 99.7% of the observations are within 3  of  See p. 181 Can ONLY be used with normal curves!

Normal Curve 34% 68% 47.5% 95% 49.85% 99.7% X-axis is standard deviations

The height of male students at CHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. a) What percent of the male students are shorter than 66 inches? b) Taller than 73.5 inches? c) Between 66 & 73.5 inches? About 2.5% About 16% About 81.5%

Chebyshev’s Rule When Distribution does NOT look like a Bell Curve Can be used with any shape of distribution Gives an “At least...” estimate

Chebyshev’s Rule The percentage of observations that are within k standard deviations of the mean is at least where k > 1 At least what percent of observations is within 2 standard deviations of the mean for any shape distribution? 75%

z score Standardized score Creates the standard normal density curve Z-score normalizes samples of data so that you can compare data variability on the same “level” (  = 0 &  = 1)

What do these z scores mean? Note: z-score is number of deviations above or below mean 2.3  below the mean 1.8  above the mean 6.1  above the mean 4.3  below the mean

Normalized Data Normalizing data allows you to compare different types of data For example: Duka is part of a pigmy colony and is 54 inches tall. The average female pigmy height is 50 inches with a σ of 1.5 in. Amy is 66 inches tall and the average height for her race is 60 inches with a σ of 3 in. Who is considered taller?

Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with µ = 45 and σ = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations below the mean. Jonathan scores 35 on this test. Does he get the job?

Mrs. Alistar read that the average amount of carrots consumed per day is 13 carrots with a standard deviation of 4. 1)This would mean that approximately 2.5% of the population consumes more than how many carrots? 2)This also means that approximately 68% of the population consume between _____ and ________ carrots.