KCP e-learning. test user - ability basic maths revision. During your training, we will need to cover some ground using statistics.
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1 During your training, we will need to cover some ground using statistics. The very mention of this word can sometimes alarm delegates who may not have done any maths or statistics since leaving school. Fear not! It is relatively painless. This Factsheet is designed to provide a refresher for those who feel they would like to get back up to speed. Arithmetic and Functional Operations You are almost certainly familiar with the four main arithmetical operations: Addition sign is + Subtraction sign is - Multiplication sign is x or * Division sign is._. or / e.g. 3._. 4 is the same as 3/4 is the same as 3 = You may be less familiar with some of the functional operations: Squaring sign is 2 This is simply multiplying a number by itself, e.g. 5 2 is the same as 5 x 5 = 25 Square Root sign is Ö This is the opposite of squaring, so, Ö 25 = 5 Throughout your RQTU training you will be able to use a calculator. To calculate a square, say, 5 2 : press 5, then the multiplication key X twice then the equals key = To calculate a square root, say, Ö25: key in the number 25, then press the Ö key
2 Symbols During the training we will use some mathematical shorthand or symbols. The main ones you will need to be familiar with are: X : this will be used for a Raw Score. A Raw Score on a test is simply the number of questions that the candidate got right. X : this is called X bar and stands for the average or mean. N : this is the number of people in a sample, so if we tested 20 graduates we could write N = 20. å : this is the capital Greek letter sigma and tells us to add something up, so, å (X - X) would be telling us to subtract the mean score from a series of raw scores then add up the results. Note here that where something is enclosed in brackets, we do whatever is inside first, then we do what the term outside the brackets tells us to do next. s : this is the lower case Greek letter sigma and is called the Standard Deviation. This is also commonly written as SD. Don t worry about what the Standard Deviation is yet - we ll cover that later. Mathematical Precedence We mentioned above that we always work out the terms within brackets first. In exactly the same way, we use an order of priority when we have an expression with multiple terms: Division and Multiplication take precedence over Addition and Subtraction So, Y = 4 + 8/(3 x 4) Here we multiply the term in the brackets first: 3 x 4 = 12 Then we divide 8 by 12 = but we round this to two decimal places: 0.67 Then we add 4 to 0.67 Therefore: Y = 4.6
3 Now, let s change our expression just a little to: Y = (4 + 8)/(3 x 4) In this case we have 4 plus 8 inside brackets So: = 12 Next we multiply the term in the second brackets: 3 x 4 = 12 Then we divide 12 by 12 = 1 Therefore: Y = 1 Measures of Central Tendency Most of us are familiar with the concept of an average. The average age of postgraduates on an English degree course at a particular university may be years. This average, or mean, is calculated by adding the ages of everybody on the course and then dividing by the number of people on the course. The formula for calculating a mean for a group of people may therefore be expressed as: = This shows that the mean (X) is equal to the sum ( å) of the raw scores (X) divided by the number of people in the group (N). The ages (in years) of 15 students on an English course are given below: 17.9, 18.0, 18.1, 18.4, 18.7, 18.8, 19.2, 19.8, 20.1, 20.5, 21.6, 28.9, 32.4, 39.1, 50.6 The mean age for the group of 15 is therefore given by: X åx N x = = The mean is the most important measure of central tendency, but there are two others which are also used. The median (or middle ) score is the point in a set of scores above which there are exactly half the scores, and below which there are the other half of the scores. In the distribution 1, 3, 5, 7, 9, 11, 13, the middle score of seven indicates the median point at which there are three scores above and three scores below. We obtain the median by arranging the scores in ascending order from the smallest to the greatest score and selecting that point above and below which there are an equal number of scores.
4 The ages (in years) of 15 students on an English Course: 17.9, 18.0, 18.1, 18.4, 18.7, 18.8, 19.2, 19.8, 20.1, 20.5, 21.6, 28.9, 32.4, 39.1, 50.6 The median age of the 15 students on the English course is 19.8 years, because there are exactly seven cases aged less than 19.8 years and seven cases aged more than 19.8 years. If the distribution (ie the set of scores) is made up of an even number of cases, the median is the mid-point between the two middle scores. The median is sometimes a useful measure of central tendency when extreme scores distort the mean. In the example above, the oldest student aged 50.6 years is much older than the others and therefore shifts the mean significantly. For some purposes, therefore, the median (19.8 years) may be a more representative measure of central tendency than the mean (24.14 years). A third measure of central tendency is the mode. The mode is the most frequently occurring score in a distribution. In the distribution 1, 1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 7, 8, 10, 11, 11, the mode is 7, because the score 7 occurs more times than any other. We might be interested in this because it could tell us what the most frequently occurring score was on an ability test.
5 The normal distribution Suppose we measured the height of every 30 year old female in England. If we plotted their heights as a frequency polygon, in other words as a chart showing how many women were found to be a particular height, the resulting chart would look something like the one on the left shown below. If we smoothed the left hand chart out, the curve would look like the picture on the right. This curve is known as the normal distribution. So, we would find that there were very few short people, most of the women would be arounf the average height, currently 1.637m (5 ft 4 1/2 in), and very few would be very tall. We can make similar assumptions in testing. On a particular test, most people will be around the average, with far fewer people at the extremes (very low scores or very high scores).
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