STATISTICS FOR PSYCH MATH REVIEW GUIDE

Transcription

1 STATISTICS FOR PSYCH MATH REVIEW GUIDE

2 ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae. Each letter in the word BEDMAS stands for a mathematical operation. When an expression involves more than one mathematical operation, they must be done in the correct order to obtain the right answer. Perform multiple operations in the following order: B - E - compute all expressions inside brackets first. ie. (3 + 4) should be simplified to 7 before performing any other operations. Note that even though the expression (4)(3) uses brackets, they refer in this case to multiplying and should not be treated as brackets in BEDMAS the second operation to be performed is to simplify any exponents in the expression ie. 5 3 = 125. D/M - the next step is to complete any division or multiplication in the question. Do these in The order (left to right) in which they appear in the question. A/S - the last step is to complete any addition or subtraction in the question. Do these in the order in which they appear in the question. In Statistics, common BEDMAS errors can result in incorrect results. For example, it is important to note that ( x) 2 means add all the x values first and then square the final answer while x 2 means square all of the individual x values first and then add the squares.

3 COMMON SYMBOLS USED IN STATISTICS Greek letters are commonly used in mathematics to denote special values. Some of the most common ones and their meanings are listed below. 4 x i i = 1 The greek symbol for upper case sigma is used to indicate that values should be summed or added. This is sigma notation. It is a short form commonly used by mathematicians to show that the indicated values should be added together. The notation shown at left means x 1 + x 2 + x 3 + x 4. µ σ σ 2 λ 5 means the sum of x 1 + x 2 + x 3 + x 4 + x 5. i x i The greek letter mu indicates the mean for an entire population of data. The greek symbol for lower case sigma indicates the standard deviation for an entire population of data. Sigma squared represents the variance for an entire population of data. Lambda. Used in Poisson distributions, lambda is the average rate of events. The mean of the distribution, µ, is equal to λt. Other symbols commonly used in statistics are: The mean for a sample from a population of data. s z The standard deviation for a sample from a population of data. The z score is the number of standard deviations a specific data value is from the mean.

4 MEASURES OF CENTRAL TENDENCY Measuring central tendency means that we are interested in finding the centre of the data (the value around which most of the data are located). There are three common measures of central tendency; the MEAN, the MEDIAN and the MODE. The calculation of each is described below. MEAN ( ) = Sum of all the data Symbolically, == The number of data MEDIAN The median is the middle number of the data when the numbers are written in order from lowest to highest. The steps to find the median are as follows: i) rewrite the data in order from lowest to highest ii) If there is an odd number of data, the median is the middle number iii) If there are and even number of data, the median is the mean of the two middle numbers (add the two middle numbers together and divide by 2). x n x MODE The mode is the number that occurs the most frequently in the list of data (remember MOST and MODE both begin with the same two letters). If there are two different numbers that occur with the same frequency, the data has two modes and is referred to as bimodal. If none of the data occur more than once, the data set has no mode. FINDING THE WEIGHTED MEAN Used when; i) some data are more significant than others (for example, if test marks count for 30% of the term mark while quizzes count for 20%) or ii)when there are multiple data with the same value (for example, a class of 25 students with a mean of 70% can be considered the same as 25 students each with a mark of 70%). In case i, if a student scores 76% on a test worth 30% (.3) of the term mark and 82% on a quiz with quizzes worth 20% (.2) of the term mark, her mean mark would be calculated as follows: weighted mean = (.3 x 76) + (.2 x 82) = Sum of the weightings In case ii, if class A has 20 students and a mean of 75% and class B has 30 students and a mean of 80%, then the weighted mean of the two classes is: weighted mean = (20 x 75) + (30 x 80) = total number of students

5 QUARTILES, DECILES & PERCENTILES Quartiles, Deciles, and Percentiles are used to measure the spread of data. QUARTILES - Divide the data into 4 equal parts. These values are denoted by Q 1, Q 2, and Q 3. Quartiles are found in a similar way to finding the median (as a matter of fact, the median is Q 2 ). To find these values; i) Re-write the data in order. ii) Find the median (refer to page 3). This is Q 2. iii) Now find the median of the first half of the data. This is Q 1. iv) Find Q 3 by finding the median of the upper half of the data. DECILES - Divide the data into 10 equal parts (D 1, D 2,...D 9 ). Find deciles in a similar way to quartiles. The third decile (D 3 ) is the number 3 10 of the way through the data (when written in order) and so on. ie. In a set of 150 numbers, find the position of D 3 by multiplying 150 x 3 = D 3 is the 45 th number when the numbers are ordered from lowest to highest (or arranged in a cumulative frequency table). PERCENTILES - Divide the data into 100 equal parts (P 1, P 2,...P 99 ). ie. P 65 is the number of the way through the data (when written in order) and so on. NOTE: P 50 = Q 2 = D 5 = median

6 MEAN DEVIATION, STANDARD DEVIATION & VARIANCE Both mean deviation and standard deviation are used to describe the spread of data about its central location. MEAN DEVIATION - A deviation is a difference. In this calculation, the deviation tells us how far each of the original numbers is from the mean of the data. Mean deviation tells us to take the mean of the deviations for all of the data. To calculate mean deviation: i) find the mean of the data ii) subtract each number of the data from the mean. If the value is negative, change it to positive (this is the absolute value of that number). iii) add all of the values (deviations) from step ii) iv) divide this answer by n (the number of data) Ex. For the data 3, 4, 5, 4, 9 i) the mean is = 5 5 ii) deviations from the mean are 5-3 = = = = = - 4 Change to 4 iii) sum of the deviations = = 8 iv) 8 = 8 = 1.6 n 5 STANDARD DEVIATION - The standard deviation (s) is calculated by; i) find the average of the squares of each of the data ( 2 x ) x ii) subtract the square of the mean of the data iii) take the square root of the answer from ii) ( nx 2 ) n VARIANCE - The variance (var) is calculated by squaring the standard deviation.

7 CREATING GRAPHS FOR STATISTICS FREQUENCY TABLE - HORIZONTAL AXIS - VERTICAL AXIS - RELATIVE FRE- QUENCY CUMULATIVE FREQUENCY - organizes data by tallying the number of times each number appears in the list of data. the bottom axis of a graph. This axis is also called the independent axis because the data displayed on it do no depend on any other variable. For example, time always appears on the horizontal axis because time will continue to move forward without being affected by any other variable. the up-and-down axis of a graph. This axis is used to display the frequency from the frequency table and is referred to as the dependent axis. ie. If you were measuring the amount of rainfall in each month of the year, the rainfall measurement would appear on the vertical axis since the amount changes depending on which month of the year is being considered. The relative frequency is calculated by dividing the frequency from the frequency table (described above) by the total number of observations. This is often converted to a percentage by multiplying the answer by 100. Cumulative frequencies are calculated by successively adding the previous frequencies in the table. Ex. Attempts at Frequency Cumulative Frequency Bar Exam INTERVAL - When creating a frequency diagram (graph), it is useful to group the data into intervals (ie , 20-30, etc). To choose intervals for a frequency diagram: i) choose to group the data into between 5 and 20 intervals. ii) make all the intervals the same length iii) choose intervals so that there are no gaps between them and so none of the data lies on an interval boundary. (ie , , etc.)

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