Chapter 1. Looking at Data-Distribution

Transcription

1 Chapter 1. Looking at Data-Distribution Statistics is the scientific discipline that provides methods to draw right conclusions: 1)Collecting the data 2)Describing the data 3)Drawing the conclusions Raw data : 1. sample data: When measurements are taken from a subset of a population, they represent sample data. 2. population data : When all indivisuals in a population are measured, the measurements represent population data. Variable : A characteristic that differs from one indivisual to the next 1. raw data from qualitative(categorical) variable : group or category name (ex) eye color (NOTE) ordinal variable(ex. tee shirt size) 2. raw data from quantitative variable: numerical values (ex. height) a) continuous b) discrete Types of data Definition : Categorical data : Individual observations are categorical response (nonnumercial, qualitative) (ex) Brand of personal computer purchased by a customer Numerical data : Individual observations are numerical response (quantitative) (ex) The number of telephone calls per day to the information desk in Scott & White hospital. (ex) height, weight Two types of numerical data: Discrete : Possible values are isolated points on the number line Continuous : The set of possible values forms an entire interval on the number line Q) What type is the above example of numerical data?

2 Distribution : A pattern of variation of a variable. Measure of Center : Mean : Average value of the sample/population. Mode : Value which occurs most frequently in the sample/population. Unimodal, Bimodal, Multimodal. Median : Middle value of the sample/population. Measure of Spread : r Percentile : For any particular number r between 0 and 100, the rth percentile is the value such that r percent of the observations in the data set fall at or below that value. Interquartile range (IQR) : Q3-Q1. Q1: lower quartile-25th percentile Q2: median -50th percentile Q3: upper quartile -75th percentile Range : max-min Sample variance : S 2 1 n n i 1 x 1 i x 2 Is the sample variance affected by outliers? Yes. Why n-1 instead n? Better estimator. Standard deviation : Square root of the variance SD is the most common way of measuring variation in a set of observations. The larger SD, the larger spread. Chebychev and Empirical Rules : Chevbychev's rule : The proportion of the observations within k standard deviations of the mean, where k 1, is at least 1-1/ k 2. 75%, 89%, and 94% of the data are within 2,3, and 4 standard deviations of the mean, respectively. Chevbychev's rule gives only a minimum proportion of the observations which lie within k standard deviation of the mean.

3 Empirical rule : If a data follows a bell- shaped curve, then approximately 68%, 95%, and 99.7% of the data are within 1,2, and 3 standard deviation of the mean, respectively. Z-scores : This quantity gives the distance between an observation and the mean expressed as a certain number of standard deviations. It is positive(negative) if the observation lies above (below) the mean. The Z score corresponding to a particular observation in a data set is observation mean Z score standard deviation The z score tells us how many standard deviations the observation is from the mean. Stem and leaf display A method of organizing quantitative data in which the stem values (leading digit(s) of the observations) are listed in a column, and the leaf (tailing digit(s)) for each observation is then listed beside the corresponding stem. Sometimes stems are repeated to stretch the display. var Percentiles Smallest 1% % % Obs 87 25% Sum of Wgt % 110 Mean Largest Std. Dev % % Variance % Skewness % Kurtosis * 5 6* 0 7* 8* * * * *

4 13* 00 14* * 0 Note : The numbers in the vertical column on the left of the display are the stem values or stems. Each number to the right of the vertical line is a leaf. Information from the SL display. 1) center of distribution. 2) overall shape of the distribution: Extent of spread : min, max, range Number and location of peaks Symmetric or skewed Frequency distribution A table that displays frequencies, and sometimes relative frequencies, for categories (categorical data), possible values (discrete data), or class intervals( continuous data) Frequency : Number of observed responses that fall into that categories. Relative frequencies: Proportion of observed responses in the categories. Histograms A picture of the information in a frequency distribution. A rectangle is drawn above each category label, possible value ( discrete data), or class interval. The rectangle' s area is proportional to the corresponding relative frequency or frequency. Information from the histogram 1) Overall shape Number of peaks: Unimodal, Bimodal... Location of center: Symmetric, skewed to the right or to the left 2) Marked deviations from the overall shape Outliers : an observation that falls outside the overall pattern of the data. Gaps: Constructing a Histogram when class widths are equal. Step1: Mark boundaries of the class intervals on a horizontal axis Step2: Use either relative frequencies or frequencies on the vertical axis. Step3: Draw the rectangle for each class directly above the corresponding interval. Constructing a Histogram when class widths are unequal. In this case frequencies or relative frequencies SHOULD NOT be used on the

5 vertical axis. Instead, the height of each rectangle, often called the density for the relative frequency of class class, is given by density=rectangle height= class width. Outliers Outliers : Observations which do not follow the regular pattern. How to find the outliers: Find the cutoff points: calculate 1.5*IQR Q1-1.5*IQR: anything lower than this will be an outlier. Q3+1.5*IQR: anything higher thatn this will be an outlier. Box-plot A five- number graphical representation for a distribution : 1. minimum. 2. first lower quartile, 25 th percentile, Q1 3. median, Q3 4. third(upper) quartile, 75 th percentile, Q3 5. maximum Quartiles cut the data in quarters. So, the lower quartile cuts off the bottom 25% of the data, the upper cuts off the top 25% of the data Note: 1. The ends of the box are at the quartiles. The length of the box(q3-q1) is called interquartile range(iqr). This box will contain 50% of the data values. 2. The median is marked by a line within the box. 3. The two vertical lines(called whiskers) outside the box extend to the smallest and largest observations within 1.5*IQR of the edges of the box. 4. Observations outside the whiskers, farther away than 1.5*IQR beyond edge of the box, are considered outliers.