Str statistics lec notes
- 1. Terms ‡ Population the totality of all possible values (measurements or counts) of a particular characteristic for specified group of objects ‡ Sample part of a population selected according to some rule or plan ‡ Parameter a descriptive property of a population ‡ Statistic any numerical value describing a characteristic of a sample ‡ Sampling the process of choosing a representative portion of a population (reading assignment: SAMPLING METHODS) ‡ Statistical Method procedure used in the collection, presentation and analysis of data STATISTICS - presentation and interpretation of chance outcomes that occur in a planned or scientific investigation - deals with other NUMERICAL DATA representing COUNTS or MEASUREMENTS or CATEGORICAL DATA that can be classified according to some criterion - looks at TRENDS in the data, patterns Uses of Statistics 1. Measures probability, predicting odds 2. For maintenance of quality use a statistic as basis or benchmark 3. For verifying claims 4. Predicting outcomes (interpolation) 5. Verifying correlations 2 Major Categories of Statistical Methods 1. DESCRIPTIVE STATISTICS collecting and describing a set of data; no inferences or conclusions about a larger set of data 2. INFERENTIAL STATISTICS analyzing a subset of data leading to predictions or inferences about the entire set of data using a sample to gauge the behaviour of the population NOTE: A statistical inference is subject to uncertainty
- 2. Introduction to Not tions £ If v e X is the v iable of inte est, and that n meas ements are taken, then the notation X1, X2, X3, ¥¤¡ ¢¡ ¢¡ ¢ ¢¦ , Xn will be used to re resent n observations. § Sigma , Indicates summation of Su ¨¨ ation Notation If variable X is the variable of interest, and that n measurements are taken, the sum of n observations can be written as THEOREMS: 1. 2. 3.
- 3. MEASURES ‡ Measures of Central Tendency ± Mean ± Median ± Mode ‡ Measures of Variability and Dis ersion © ± Range ± Average deviation ± Variance ± Standard deviation Measures of Central Tendency MEAN ‡ The sum of all values of the observations divided by the total number of observations ‡ The sum of all scores divided by the total fre uency Properties ‡ The most stable measure of central tendency ‡ Can be affected by extreme values ‡ Its value may not be an actual value in the data set ‡ If a constant c is added/substracted to all values, the new mean will increase/decrease by the same amount c MEDIAN ‡ Positional middle of an array of data ‡ Divides ranked values into halves with 50% larger than and 50% smaller than the median value. Properties ‡ The median is a positional measure ‡ Can be determined only if arranged in order ‡ Its value may not be an actual value in the data set ‡ It is affected by the position of items in the series but not by the value of each item ‡ Affected less by extreme values
- 4. MODE ‡ Value that occurs most fre uently in the data set ‡ Locates the point where scores occur with the greatest density ‡ Less popular compared to mean and median measures Properties ‡ It may not exist, or if it does, it may not be unique ‡ Not affected by extreme values ‡ Applicable for both qualitative and quantitative data Measures of Variability and Dispersion RANGE ‡ Measure of distance along the number line over where data exists ‡ Exclusive and inclusive range ± Exclusive range = largest score - smallest score ± Inclusive range = upper limit - lower limit Properties ‡ Rough and general measure of dispersion ‡ Largest and smallest extreme values determine the range ‡ Does not describe distribution of values within the upper and lower extremes ‡ Does not depend on number of data ABSOLUTE DEVIATION Average of absolute deviations of scores from the mean (Mean Deviation) or the median (Median Absolute Deviation) Properties ‡ Measures variability of values in the data set ‡ Indicates how compact the group is on a certain measure VARIANCE ‡ Average of the square of deviations measured from the mean ‡ Population variance ( 2) and sample variance (s2)
- 5. Properties ‡ Addition/subtraction of a constant c to each score will not change the variance of the scores ‡ Multiplying each score by a constant c changes the variance, resulting in a new variance multiplied by c2 STANDARD DEVIATION ‡ Square root of the average of the square of deviations measured from the mean square root of the variance ‡ Population standard deviation ( ) and sample standard deviation (s) Why n-1? ‡ Degrees of freedom ± Measure of how much precision an estimate of variation has ± General rule is that the degrees of freedom decrease as moreparameters have to be estimated ‡ Xbar estimates ‡ Using an estimated mean to find the standard deviation causes the loss of ONE degree of freedom Properties ‡ Most used measure of variability ‡ Affected by every value of every observation ‡ Less affected by fluctuations and extreme values ‡ Addition/subtraction of a constant c to each score will not change the standard of the scores ‡ Multiplying each score by a constant c changes the standard deviation, resulting in a new standard deviation multiplied by c CHOOSING A MEASURE ‡ Range ± Data are too little or scattered to justify more precise and laborious measures ± Need to know only the total spread of scores ‡ Absolute Deviation ± Find and weigh deviations from the mean/median ± Extreme values unduly skews the standarddeviation ‡ Standard Deviation ± Need a measure with the best stability ± Effect of extreme values have been deemed acceptable ± Compare and correlate with other data sets