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## Lies, damned lies & statistics

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**Lies, damned lies & statistics**Communication Research week 10**Basics of descriptive statistics**• Statisticians use mathematical methods to analyse, summarise and interpret data that have been collected • Descriptive statistics describe the basic features of the study and allows the researcher to get a feel for the data • The choice of statistical method of analysis depends on the data that have to be analysed Communication Research Spring 2005**Descriptive vs inferential statistics**• Descriptive statistics refer to methods used to obtain, from raw data, information that characterises or summarises the whole set of data • Inferential statistics allow us to generalise from the data collected to the general population they were taken from Communication Research Spring 2005**Different statistical measures**• Raw data is unorganised but can be tabulated to make it easier to understand and to interpret • It is usually presented as a frequency table or graph • A frequency chart will allow a researcher to see trends or groupings of data and how they are distributed Communication Research Spring 2005**Some Basic Concepts Related to Statistics**Data: The raw material of statistics. Numbers that result from measurements or counting. Statistics: The field of study concerned with the collection, organization, summarization and analysis of data and the drawing of inferences about a body of data when only a part of the data is observed. Sources of Data Routinely kept records Surveys Experiments External Sources Communication Research Spring 2005**Some Basic Concepts Related to Statistics**Random Variable:A variable whose values arise as a result of chance factors and cannot be exactly predicted in advance. Population:A population of entities is defined as the largest collection of entities for which we have an interest at a particular time. Sample:A part of a population. Communication Research Spring 2005**The Simple Random Sample**Statistical Inference The procedure by which we reach a conclusion about a population on the basis of the information contained in a sample that has been drawn from that population. Simple random sample If a sample of size n is drawn from a population of size N in such a way that every possible sample of size n has the same chance of being selected, the sample is called a simple random sample. 2 out of 4=2C4=6 Communication Research Spring 2005**Characteristics of each distribution**• Location – where on the axis is the distribution positioned? • Dispersion – how broad is the distribution? • Shape – what is the form (appearance, pattern) of the distribution? • The type of data you have to analyse will determine the statistical measure chosen • Statistics describing the location of the distribution are called measures of central tendency Communication Research Spring 2005**Measures of central tendency – the mean**• The mean is the sum of all observed data values divided by the sample size (the arithmetic average) • Describing data that are interval or ratio in nature (eg speed of response, age in years) calls for the mean • One of the main disadvantages is that it is most profoundly affected by extreme scores Communication Research Spring 2005**Calculating a Mean Score**Scores: 79 81 82 86 86 88 91 93 95 97 total = 878 Divide by n = 10 scores Mean = 87.8 Communication Research Spring 2005**Measures of central tendency – the median**• The median is the score or the point of distribution above which one half of the scores lie eg in a simple set of scores such as 1, 3, & 5 the median is 3 • The median is best suited to data that are ordinal or ranked ( eg birth order, rank in class) • To compute the median • Order the scores from lowest to highest • Count the number of scores • Select the middle score • When the number of scores is even, find the mean of the two middle scores • eg 31 33 35 38 40 41 42 43 44 46 47 48 49 50 • N = 14 (no of scores); Median = (42 + 43) ÷ 2 = 42.5 Communication Research Spring 2005**24**24 25 25 26 26 Mean = 25 Range = 3 16 19 22 25 28 30 35 Mean = 25 Range = 20 Two distributions of scoresDistribution 1 Distribution 2 Communication Research Spring 2005**Measures of central tendency – the mode**• The mode is the most frequently observed value in the frequency distribution ie it is the score that occurs most frequently • The mode is best used for nominal data and for data that are qualitative in nature such as gender, eye colour, ethnicity, school or group membership • In the following list of numbers: • 58 27 24 41 27 26 41 53 24 29 41 53 47 28 56 • The mode is 41 because it occurs 3 times • A common mistake is to identify the mode as how frequently the value occurs (3) not the value itself (41) Communication Research Spring 2005**Which measure when?**Communication Research Spring 2005**Three Measures of Variability**• Range: the difference between the highest and lowest scores in a distribution of scores. • Variance: a measure of dispersion indicating the degree to which scores cluster around the mean score. • Standard deviation: index of the amount of variation in a distribution of scores. Communication Research Spring 2005**Standard deviation**• SD is a measure of the variability indicating the degree to which all observed values deviate from the mean • SD can only be used for interval and ratio data • It is the most frequently used statistic as a measure of dispersion or variability • The larger the SD, the more variable the set of scores is Communication Research Spring 2005**COMPUTING DEVIATION SCORES**Raw Mean DEV. SQUARED score score deviation score 4 - 10 = -6 36 8 - 10 = -2 4 9 - 10 = -1 1 10 - 10 = 0 0 10 - 10 = 0 0 10 - 10 = 0 0 12 - 10 = 2 4 13 - 10 = 3 9 14 - 10 = 4 16 90/9 = 10.00 = MEAN 70/9 = 7.77 = Variance STANDARD DEVIATION: (Square Root of Variance) = 2.79 Communication Research Spring 2005**Types of Variables**• Variable • Element that is identified in the hypothesis or research question • Property or characteristic of people or things that varies in quality or magnitude • Must have two or more levels • Must be identified as independent or dependent Communication Research Spring 2005**Independent Variables**• Manipulation or variation of this variable is the cause of change in other variables • Technically, independent variable is the term reserved for experimental studies • Also called antecedent variable, experimental variable, treatment variable, causal variable, predictor variable Communication Research Spring 2005**Dependent Variables**• The variable of primary interest • Research question/hypothesis describes, explains, or predicts changes in it • The variable that is influenced or changed by the independent variable • In non-experimental research, also called criterion variable, outcome variable Communication Research Spring 2005**Relationship Between Independent and Dependent Variables**• Cannot specify independent variables without specifying dependent variables • Number of independent and dependent variables depends on the nature and complexity of the study • The number and type of variables dictates which statistical test will be used Communication Research Spring 2005**Issues of Reliability and Validity**• Reliability = consistency in procedures and in reactions of participants • Validity = truth - Does it measure what it intended to measure? • When reliability and validity are achieved, data are free from systematic errors Communication Research Spring 2005**Threats to Reliability and Validity**• If measuring device cannot make fine distinctions • If measuring device cannot capture people/things that differ • When attempting to measure something irrelevant or unknown to respondent • Can measuring device really capture the phenomenon? Communication Research Spring 2005**Other Sources of Variation**• Variation must represent true differences • Other sources of variation • Factors not measured • Personal factors • Differences in situational factors • Differences in research administration • Number of items measured • Unclear measuring device • Mechanical or procedural issues • Statistical processing of data Communication Research Spring 2005**Types of variables**Communication Research Spring 2005**Definitions**• Variable: a characteristic that changes or varies over time and/or different subjects under consideration. • Changing over time • Blood pressure, height, weight • Changing across a population • gender, race/ethnicity Communication Research Spring 2005**Definitions (con’t)**• Quantitative variables (numeric):measure a numerical quantity of amount on each experimental unit • Qualitative variables (categorical):measure a non numeric quality or characteristic on each experimental unity by classifying each subject into a category Communication Research Spring 2005**Categorical variables**• Nominal:unordered categories • Race/ethnicity • Gender • Ordinal:ordered categories • likert scales( disagree, neutral, agree ) • Income categories Communication Research Spring 2005**Univariate statistics (numerical variables)**• Summary measures • Measures of location • Measures of spread • Overall pattern (distribution) • Unimodal (one major peak) vs. bimodal) (2 peaks) • Symmetric vs. skewed • Outliers-an individual value that falls outside the overall pattern Communication Research Spring 2005**Skewness**• The skewness of a distribution is measured by comparing the relative positions of the mean, median and mode. • Distribution is symmetrical • Mean = Median = Mode • Distribution skewed right • Median lies between mode and mean, and mode is less than mean • Distribution skewed left • Median lies between mode and mean, and mode is greater than mean Communication Research Spring 2005**Relative positions of the mean and median for (a)**right-skewed, (b) symmetric, and(c) left-skewed distributions Note: The mean assumes that the data is normally distributed. If this is not the case it is better to report the median as the measure of location. Communication Research Spring 2005**Summary statisticsMeasures of spread (scale)**• Variance: The average of the squared deviations of each sample value from the sample mean, except that instead of dividing the sum of the squared deviations by the sample size N, the sum is divided by N-1. • Standard deviation: The square root of the sample variance • Range: the difference between the maximum and minimum values in the sample. Communication Research Spring 2005**Normal curvessame mean but different standard deviation**Communication Research Spring 2005**Graphical display of numerical variables (histogram)**Class Interval Frequency 20-under 30 6 30-under 40 18 40-under 50 11 50-under 60 11 60-under 70 3 70-under 80 1 Communication Research Spring 2005**86**77 91 60 55 76 92 47 88 67 23 59 72 75 83 77 68 82 97 89 81 75 74 39 67 79 83 70 78 91 68 49 56 94 81 Graphical display of numerical variables (stem and leaf plot) Stem Leaf Raw Data 2 3 4 5 6 7 8 9 3 9 7 9 5 6 9 0 7 7 8 8 0 2 4 5 5 6 7 7 8 9 1 1 2 3 3 6 8 9 1 1 2 4 7 Communication Research Spring 2005**S < 0**S > 0 S = 0 Symmetric (Not Skewed) Positively Skewed Negatively Skewed Graphical display of numerical variables (box plot) Communication Research Spring 2005**Univariate statistics(categorical variables)**• Summary measures • Count=frequency • Percent=frequency/total sample • The distribution of a categorical variable lists the categories and gives either a count or a percent of individuals who fall in each category Communication Research Spring 2005**Displaying categorical variables**Communication Research Spring 2005**Common Applications**• T-Tests – the independent t-test is used to test for a difference between two independent groups (like males and females) on the means of a continuous variablecontinuous variable. • one sample – compare a group to a known value • For example, comparing the IQ of convicted felons to the known average of 100) • paired samples – compare one group at two points in time • For example, comparing pretest and posttest scores • independent samples – compare two groups to each other Communication Research Spring 2005**Common Applications**• The Pearson's correlation is used to find a correlation between at least two continuous variables. The value for a Pearson's can fall between 0.00 (no correlation) and 1.00 (perfect correlation). • Other factors such as group size will determine if the correlation is significant. Generally, correlations above 0.80 are considered pretty high Communication Research Spring 2005**60**50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Days Absent Non-significant t-test Common Applications Male Female Number of people Communication Research Spring 2005**60**50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Days Absent Significant t-test Common Applications Male Female Number of people Communication Research Spring 2005