Descriptive Statistics C H A P T E R 5 P P

Transcription

1 Descriptive Statistics C H A P T E R 5 P P

2 Graphing data Frequency distributions Bar graphs Qualitative variable (categories) Bars don t touch Histograms Frequency polygons Quantitative variable (ordinal, interval, or ratio scale) Others: Pie chart Stem and leaf Scatterplot

3 Example grade distribution Class interval frequency distribution A 1 A- 2 B+ 3 B 7 B- 8 C+ 6 C 3 C- 2 D 1 F 1 N = 34

4 Number per 100,000 population Graphs! Read X and Y axis carefully Death Rates in America Age 1-4 Age Year

5 Example bar graph Rauscher, Shaw, & Ky (1993). Mozart Effect N = 36 college students

6 Lots of cool graphs! Florence Nightingale s coxcomb diagram Blue: died of sickness; Red: died of wounds; Black: died of other causes

7 Graph interpretation Careful to read values on each axis graphs can be deceiving! Reminiscence bump Recency effect

8 Descriptive statistics Data collected in a study = raw data Reports of a study = summary data Descriptive statistics provide that summary Measures of central tendency Describe middleness of distribution of scores Mean Median Mode Measures of variation Describe width or dispersion of a distribution Range Standard deviation Variance

9 Descriptive statistics Measure of central tendency Mean Mean for population = sum of scores # of scores in distribution μ = ΣX N Mean for sample = sum of scores # scores in distribution M or X = ΣX N

10 Mean as the balance point The mean balances the distances (or deviations) of all scores Scores (x) X = 20 N = 4 M = 5 Mean Distance from mean X = 0

11 Effect of changing 1 score X = X / N = M = / X = X / N = M = / The mean is not a robust statistic It is highly influenced by a single outlier score

12 Adding a constant X / / If you add, subtract, multiply or divide all scores by constant: The same change is made to M

13 Descriptive statistics Measure of central tendency Mean Mean for population = sum of scores # of scores in distribution µ = X / N Mean for sample = sum of scores # scores in distribution X or M = X / N Median Middle score in distribution Order scores from highest to lowest If N is even number, average the two middle scores

14 Calculating the median for RTs scores Median Mean sorted Add Hi X Add Lo X Median is a robust statistic!

15 Descriptive statistics Measure of central tendency Mean Mean for population = sum of scores / # of scores in distribution µ = X / N Mean for sample = sum of scores / # scores in distribution X or M = X / N Median Middle score in distribution Order scores from highest to lowest If N is even number, average the two middle scores Mode Score that occurs with greatest frequency

16 Example grade distribution A 1 A- 2 B+ 3 B 7 B- 8 C+ 6 C 3 C- 2 D 1 F 1 N = 34 M = Median = 81 Mode = B-

17 Can have 2+ modes Sample grade distribution with 2 modes A A- B+ B B- C+ C C- D F

18 Types of distributions Normal distribution Bell-shaped Symmetrical Only 1 mode Mean, median, mode all equal Kurtosis: spread of distribution How flat or peaked Mesokurtic: medium peak (like normal distribution) Leptokurtic: tall and thin Platykurtic: flat and broad

19 Measures of central tendency Indicators of the shape of the distribution How mean, median, and mode change w/ shape of distribution Normal distribution Positive skew Tail to positive scores Negative skew Tail to negative scores Positive skew Negative skew

20 Which measure of central tendency to use? If interval or ratio data and normally-distributed Use mean If interval or ratio data and there are outliers or a skewed-distribution Use median If nominal data Use mode But, that s not enough info

21 Measures of variation Range Difference between lowest and highest scores in a distribution = Maximum score minimum score Easily distorted by an outlier (low or high score) Standard deviation Average distance of scores in a distribution from the mean If sum deviations from mean = zero! SO Average deviation: Use absolute values Standard deviation: Use squared deviation scores For population: σ = Σ(X μ)2 N

22 Example grade distribution A 1 A- 2 B+ 3 B 7 B- 8 C+ 6 C 3 C- 2 D 1 F 1 N = 34 M = Median = 81 Mode = B- s = 7.92 M - s = 72.5 M = M + s = 88.3 Note: most scores are w/in 8 pts of mean

23 Calculating standard deviation (σ) 1. Calculate deviation score (score mean) 2. Square deviations 3. Sum squared deviations 4. Divide by N 1. N = # of scores 2. This step = variance 5. Take square root of value RTs x - M (x - M) 2 Avg = sum of (X-M) Variance: sum divided by N SD: square root of sum/n

24 Calculating standard deviation (s) 1. Calculate deviation score (score mean) 2. Square deviations 3. Sum squared deviations 4. Divide by N or N This step = variance 2. Use N for population 3. Use N-1 to estimate population from sample 5. Take square root of value RTs x - M (x - M) 2 Avg = sum of (X-M) 2 sd = Variance: sum divided by N = SD: square root of sum/n-1

25 Measures of variation Standard deviation of population σ = Σ(X μ)2 N Standard deviation of sample (when estimating population) s = Variance Σ(X M)2 N 1 Population = σ 2 = Σ(X μ)2 N or sample = s 2 = Σ(X μ)2 N

26 Why use N 1? Sample is less variable than the population Divide by smaller # so yields more conservative estimate of variance or SD Makes variance score larger Use n-1 so can make conclusions about population (not just describe your sample)

27 Thank you, Excel! For example, if data is in column B from row 1 to 20 Sum: =sum(b1:b20) Mean: =average(b1:b20) Median: =median(b1:b20) Mode: =mode(b1:b20) Maximum score: =max(b1:b20) Minimum score: =min(b1:b20) Range: Subtract Max score from Min score