GCSE HIGHER Statistics Key Facts

Transcription

1 GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information you need! 2. the question is brief, clear and is not biased or leading in any way 3. you include a timeframe in the question if appropriate (per week, per month, etc.) 4. you include response options that cover all possible answers, with no gaps or overlaps Pilot Survey a small scale test survey done to check whether a full survey will work (to check if questions are appropriate and can be understood etc) Control Group a randomly selected group that are not subject to any test procedures being used in an experiment (used to compare with results from those under test) Capture/Recapture Method used to estimate the size of a population Questionnaires vs. Face to Face Interviews Questionnaire Interview Advantages Less expensive Quicker Interviewer can explain questions and/or gauge responses Disadvantages Can reach more people Can be less embarrassing Number of responses is unpredictable All questions may not be answered More likely to get answers Expensive Time consuming Interviewer may influence answers Interviewee may not be honest

2 SAMPLING Population everybody or everything that could be involved in an investigation Census collecting data from every member of the population Sample collect data from part of the population Sampling Frame a list of all the people or items in the population Sampling Unit a person or item in the population How to take a RANDOM SAMPLE: 1. give every item in the sampling frame a number 2. use a random number generator to generate a quantity of numbers, according to the sample size 3. select each piece of data allocated with each randomly generated number How to take a STRATIFIED SAMPLE: - divide your sample up, in proportion to the different group sizes within your population: Group size Total population size x sample size

3 Central Tendency (Average) and Dispersion (Spread) For a set of n items of discrete data: the median (Q 2 ) is the (n + 1) th 2 piece of data the lower quartile (Q 1 ) is the (n + 1) th 4 piece of data the upper quartile (Q 3 ) is the 3 (n + 1) th 4 piece of data Interquartile Range (IQR) = Upper Quartile Lower Quartile Mean = Σx (the sum of the data items number of pieces of data) n For a frequency distribution (data in a table, including grouped continuous data): When finding MEAN FROM A TABLE, remember to MULTIPLY the data (x) by the frequency (f) *for GROUPED DATA, find the MID-POINTS first (then, mean = sum of data sum of frequencies) (this is on the exam paper formula sheet) (for grouped data, x represents the mid-points of the groups) For grouped data, the group (or class) that contains the median is the group that has the (n + 1) th piece of data in it 2

4 To find an estimate for the Median and IQR for a grouped frequency distribution, you need to first draw a CUMULATIVE FREQUENCY CURVE: Median (Q2) = ½ (read off from halfway up the cumulative frequency axis) Lower Quartile (Q1) = ¼ (read off from ¼ of the way up the cumulative frequency axis) Upper Quartile (Q3) = ¾ (read off from ¾ of the way up the cumulative frequency axis) Interquartile Range (IQR) = Q3 - Q1 Medians and Quartiles can be represented on Box Plots, which also show Data Distribution and Skew: For skewed data, an OUTLIER is defined as a piece of data that is: Less than LQ x IQR or More than UQ x IQR

5 When asked to COMPARE data or graphs, you need to compare an average (MEAN OR MEDIAN) and a measure of dispersion (INTERQUARTILE RANGE or STANDARD DEVIATION) (this is on the exam paper formula sheet) NORMALLY DISTRIBUTED DATA When represented in a histogram, normally distributed data is symmetrical about the mean the data is said to follow a NORMAL DISTRIBUTION For a Normal Distribution, Mean = Median = Mode 99.8% of the data is within 3 standard deviations either side of the mean 95% of the data is within 2 standard deviations either side of the mean 68% of the data is within 1 standard deviation either side of the mean For normally distributed data, an OUTLIER is a piece of data that is more than 3 standard deviations away from the mean

6 Standardised Scores useful for comparing exam marks in two different subjects Standardised Score = score - mean standard deviation Standard Deviation can be estimated from a normal distribution curve remember, the difference between the mean value and the highest or lowest value is 3 standard deviations Curve A: Mean = 15 S.D. = 10 3 = 3.3 Curve B: Mean = 20 S.D. = 15 3 = 5 A HISTOGRAM is also a good visual representation of any SKEW: Positive Skew (majority of data at lower end) Negative Skew (majority of data at lower end) HISTOGRAMS with UNEQUAL CLASS INTERVALS remember that the AREA of the bar represents the frequency, so: HEIGHT OF BAR = FREQUENCY CLASS WIDTH

7 Scatter Graphs and Correlation When asked to describe correlation on a scatter graph, you need to say whether the correlation is positive or negative (or if there is no correlation) When asked to interpret correlation, you need to explain what the correlation means: Positive correlation means that as one variable increases, so does the other Negative correlation means that as one variable increases, the other decreases The strength of the correlation can be determined by calculation Spearman s Rank Correlation Coefficient, using this formula (which is given on the exam paper formula sheet): To calculate r s, you need to put each of the two variables into rank order and then d is the difference between the ranks for each data pair. n = the number of pairs of data. Equation of a line of best fit

8 Time Series and Seasonal Variation 1. Plot the data from this table onto a Time Series Graph 2. Calculate moving averages For this data, 4-point moving averages are sensible because the years are split into quarters First 4-point moving average Second 4-point moving average = = 28 = = 27 etc Plot the moving averages onto the time series graph 4. Draw a trend line through the moving averages 5. The seasonal variations can then be calculated by finding the difference between the actual value and the trend line value at a particular point in time. The seasonal variations for quarter 3 in 2007 and 2008 are indicated on this graph

9 Mean Seasonal Variation and Making Predictions PROBABILITY OR means ADD the probabilities (OR +) AND means MULTIPLY the probabilities (AND x) (p + q) 3 = p 3 + 3p 2 q + 3pq 2 + q 3 (p + q) 4 = p 4 + 4p 3 q + 6p 2 q 2 + 4pq 3 + q 4 (p + q) 5 = p 5 + 5p 4 q + 10p 3 q p 2 q 3 + 5pq 4 + q 5 Using Pascal s Triangle to help:

10 can you complete the expansion for (p + q) 3 and write out the expansion for (p + q) 4 : (p + q) 1 = p + q (p + q) 2 = p 2 + 2pq + q 2 (p + q) 3 = p 3 + 3p 2 q + (p + q) 4 = If p represents the probability of success and q represents the probability of failure, then apply the binomial distribution to solving this problem: The probability that a seed produces flowers when it is planted is 75%. Four seeds are planted. (p + q) 1 = p + q (p + q) 2 = p 2 + 2pq + q 2 (p + q) 3 = p 3 + 3p 2 q + 3pq 2 + q 3 The probabilities for the events when n dice are rolled will be the terms in the expansion of (p + q) n The binomial distribution with n trials and probability of success p is denoted by B(n, p)