Lecture 2. Descriptive Statistics: Measures of Center

Transcription

1 Lecture 2. Descriptive Statistics: Measures of Center Descriptive Statistics summarize or describe the important characteristics of a known set of data Inferential Statistics use sample data to make inferences (or generalizations) about a population

2 Numerical Summaries: Measuring Center of the Data Set Example. You want to buy a 3-4 bdrm house. Need info on real estate sales in Reno, say on 200 recently sold 3-4bdrm houses. Data: $325,300, $287,650, $589,900, $230,900,, $455,800. Q: What is the average selling price for a 3-4 bdrm house? What does AVERAGE mean? Most common? Most frequent? Mode Dividing selling prices in half, i.e. half are lower and higher that the average? Median Arithmetic average of all selling prices. Mean

3 MEAN POPULATION MEAN: μ Sample: n=sample size SAMPLE MEAN x =(sum of all observations)/(the total number of observations) where x = = (sum of all observations)/sample size n x. is the sum of all n observations. Example: Data with 5 observations of quiz scores: 8, 5, 7, 3, 7. n=5. Mean score = ( )/5=6.

4 MEDIAN SAMPLE MEDIAN is the middle value when the data is arranged in an increasing (or decreasing) order. Equal numbers of observations are larger and smaller than median. SORT the data, then Odd number of observations - the median is the middle observation. Even number of observations - the median is the average of the two middle values. Example. Quiz scores: 8, 5, 7, 3, 7. Find median of the quiz scores. Step 1. Sort the data: 3, 5, 7, 7, 8. Step 2. Even or odd n? Odd. Step 3. Median is the middle observation =7. Let s add an observation: New quiz data: 8, 5, 7, 6, 3, 7. Find median of the quiz scores. Step 1. Sort the data: 3, 5, 6, 7, 7, 8. Step 2. Even or odd n? Even. Step 3. Median is the average of the two middle observations. (6+7)/2=6.5=median.

5 MODE SAMPLE MODE is the most frequent value in the data set. Example. Quiz scores: 8, 5, 7, 3, 7. Find mode of the quiz scores. Answer: Mode is 7 because 7 is most frequent. Note that mode may not always be unique. Why? Example: 1,2,4, 1, 5, 2, 7. Mode: 1 and 2: bimodal data Note, that mode does not always exists. Example: 1, 2, 5, 7, -1, 9, 3. No mode, all observations are different.

6 Midrange Midrange is the value midway between the maximum and minimum values in the original data set Midrange = maximum value + minimum value 2 Example. Quiz scores: 8, 5, 7, 3, 7. Find midrange of the quiz scores. Solution: Maximum score=8, minimum score=3, midrange=(8+3)/2=11/2=5.5

7 Round-off Rule for Measures of Center Carry one more decimal place than is present in the original set of values. Example. Quiz scores: 8.3, 5, 7, 3, 7. reported with no decimals. Find mean. Solution: mean=( )/5=

8 Mean from a Frequency Distribution Assume that in each class, all sample values are equal to the class midpoint, and use class midpoints of classes for variable x, f=frequency. Example: class freq Mid point fx 1 _ _ Mean=(30+48)/(5+3)=78/8= 9.75

9 OUTLIERS OUTLIERS observations FAR outside the regular pattern of the data. Where do outliers come from? Errors of measurements or recording in those cases, people tend to disregard them. Natural order of things they point to very important phenomena like floods, heat waves, hurricanes, etc. Should not be discarded but studied. Example. Waiting times (in minutes) for a bus, 100 observations waiting time, min

10 OUTLIERS AND MEASURES OF CENTER Example. Take quiz scores. 3, 5, 7, 7, 8. Suppose I made a recording error and instead of 8 recorded 88. New data: 3, 5, 7, 7, 88. New median= 7 = old median NO change, New mode = 7 = old mode NO change, New mean = ( )/5=22 LARGE change, old mean=6. MEDIAN AND MODE are RESISTANT (ROBUST) TO OUTLIERS i.e. do not change if we add outliers to a data set. MEAN IS SENSITIVE TO OUTLIERS i.e. changes if we add an outlier to a data set. Summary: If you do not want the very large or the very small (outliers) observations to affect the information you are getting about the center of the data ask for median rather than the mean.

11 Best Measure of Center

12 Symmetry and Skewness of a Distribution Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half Symmetric histogram Skewed distribution of data is skewed if it is not symmetric and if it extends more to one side than the other Skewed histograms

13 Skewness MEAN = MEDIAN MEAN < MEDIAN MEAN > MEDIAN

14 MEASURES OF SPREAD (VARIABILITY, VARIATION, DISPERSION) Variability in Nature, life, and various processes we investigate is fundamental to the theory of Statistics A B

15 Measures of variability Range and Standard Deviation RANGE is the difference between the largest and the smallest observations. Range = maximum value minimum value The more variability or spread is in the data, the larger the difference between the min and max, the larger the range. Example. For the two data sets summarized in the histograms: A: min = 6.96, max = 13, range= = 6.04 B: min = , max = 40.93, range= (-22.74)=63.67

16 Sample Standard Deviation STANDARD DEVIATION measures average deviation of the data from the mean. s = Σ (x - x) 2 n - 1 Sample Standard Deviation (Shortcut Formula) s = nσ(x 2 ) - (Σx) 2 n (n - 1)

17 Computing standard deviation Example. Data is number of km to school for a sample of 18 kids Mean=(sum of obs)/18=92/ Numerator for Variance=(2-5.1) 2 + (5-5.1) (1-5.1) Denominator = n-1= 17 Variance= 7.87 Standard deviation= square root of km New data set: New variance and standard deviation: variance = 40.57, st. deviation= 6.37 km. Big change in standard deviation, caused by the presence of an observation far from the rest of the data, an outlier.

18 Standard Deviation - Properties The standard deviation is a measure of average variation of all values from the mean. The value of the standard deviation s is usually positive, always nonnegative. Sample variance = 0 only if all deviations from the mean are zero, that is all observations are the same. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). The units of the standard deviation s are the same as the units of the original data values.

19 Variance The variance of a set of values is a measure of equal to the square of the standard deviation. variation Sample variance: Square of the sample standard deviation s Population variance: Square of the population standard deviation σ

20 Round-off Rule for Measures of Variation Carry one more decimal place than is present in the original set of data. If possible, round only the final answer, not values in the middle of a calculation.

21 Empirical ( ) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean.

22 The Empirical Rule

23 Empirical Rule at work Example: Exam scores follow an approximately bell shaped distribution with mean 70 and standard deviation 5. What is the approximate percentage of students scoring: a. 2 st. dev. or higher above the mean? b. At most 3 st. dev. below the mean? c. Between 2 st. dev below and 1 st. dev. above the mean? d. More than 4 st. dev. above the mean? Solution. a. 2.4% + 0.1% = 2.5% b. 0.1% c.13.5%+68%=79.5% d. 0%

24 Definition The coefficient of variation (or CV) for a set of sample or population data, expressed as a percent, describes the standard deviation relative to the mean. Sample Population CV = s 100% x CV = σ 100% µ Example: For the distance to school data (no outlier) CV=5.1/2.81=1.81 With outlier: CV= 6.72/6.37=1.05