STATISTICS 1

Topic 1: Mathematical Models and Topic 2: Locationmathematical model=simplification of real world situationnumerical data- quantitative; non numerical data- qualitativecontinuous variable takes any value in given range, discrete can only take specific valuesGROUPED DATA- useful to group data when in large amountcontinuous data can be shown in more than one way,eg. class weight= 31-40, class boundary=30.5 - 40.5, class width=10(40.5-30.5)MEAN- find mean of 3, 10, 15, 7, 8= 43/8= 8.6 *now add 3 to all values and find mean* = 58/5 = 11.6---11.6 = 8.6+ 3---THIS IS CALLED CODING - makes values easier to work with !!eg. number b is subtracted/added from each value, then mean is same w b added/subtractedalso works with division+multiplication!!eg x=10y +50if x =60, then y=60-50 / 10 = 1(code all data and work out mean of y, then use coding to find mean of x)Topic 3: Measures of DispersionSTANDARD DEVIATION: essentially, SD is measure of how close values are to the mean3 commonly used measures of spread: the range, IQR, standard deviationbased on all values of data- **v sensitive to outliers**variance= (standard deviation)² VARIANCE= E(x1-mean)squared / n OR Exi sq/n -(mean)sq*******SD= sqr root (variance)IN MOST DISTRIBUTIONS 67% data lie within 1 sd from mean, most in 2 sd. Values that lie >2 sd from mean are considered OUTLIERS - should be treated carefully - MEAN + STANDARD DEVIATIONif you add/ subtract c to each piece of data, sd remains unchanged. if you multiply/ divide every value by c then the sd is multiplied/ divided by ceg, data set coded by x=y x 10- standard deviation of y= 1.41,***therefore sd of x = 1.41 x 10 = 14.17Topic 4: Representation of DataQUARTILES AND BOX PLOTS- ** smaller the range, more consistent the data is**data that is abnormally small/ large is classified as an OUTLIER. - less than LQ- 1.5 x IQR more than UQ +1.5 x IQRSHAPES OF DISTRIBUTION: symmetrical/ no correlation IF Q2-Q1= Q3-Q2positively skewed if Q2 - Q1 Q3 - Q2negatively skewed if Q2 - Q1 > Q3 - Q2 HISTOGRAMS: used to display grouped continuous data, important points to remember: *Area of each bar in histogram should be in proportion to the frequency *when class width not all equal, proportional areas can be achieved by plotting frequency densities on vertical areas where FREQUENCY DENSITY = FREQUENCY / CLASS WIDTH rules for plotting histogram: 1.plot frequency densities on y axis 2. choose sensible scales 3. label both axis 4. give it a title! AREA IS PROPORTIONAL TO FREQUENCY A= K x FTopic 5: Probability possible things that can happen are called outcomes. eg outcome of throwing coin= heads/ tails ---->outcomes can be shown by: list/ sample space/ tree diagram for equally likely outcomes the probability of event X occurring is P(x)= number of possible outcomes/ total number of outcomes Eg. p of picking vowel from 'PROBABILITY' = 4/11 ***************************************** when outcomes not equally likely we can use relative frequency to estimate probabilities data taken from experiment/ observation Eg, coin thrown 50 times + lands heads 30 times, so P(tails) = 20/50 OR 2/5***************************************************************SAMPLE SPACEEG. P(more than 5)= 26/362. P(less than 10) = 30/36