Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS

Transcription

1 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3- Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS A) Frequency Distributions For Samples Defn: A FREQUENCY DISTRIBUTION is a tabular or graphical display of the frequencies or number of occurrences of the values of a variable in the data set. EXAMPLE A DDT contamination study was done on a portion of the Tennessee River in 98. The experiment involved sampling five fish at each of several locations along the river. For each fish, the variables measured were: location (miles upstream of the mouth of the river), species, length (centimeters), weight (grams), and DDT concentration in the fillet (ppm). Part of the data set follows. Obs Location Species Length Weight Concentration 275 catfish catfish catfish catfish catfish buffalofish buffalofish bass bass bass bass catfish

2 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-2 ) Summarizing Categorical Data One-way Frequency Table Species Absolute Frequency Relative Frequency Catfish 6 6/2 =.5 Buffalofish 2 2/2 =.67 Bass 4 4/2 =.333 TOTAL 2. Bar Chart Count bass buffalofish catfish SPECIES 2) Summarizing Continuous Data Dot Plot: each dot represents one observation in the dataset Example: Fish Lengths

3 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-3 _ _ Length (cm) Stem-and-Leaf Plots: each dot includes the data value as part of the graphic e.g. DDT (ppm) in 4 fish collected in the Tenn. R. Stem Leaf Count

4 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-4 To construct a stem-and-leaf plot: ) find the max and min values in the dataset 2) decide which digits in a value are significant ( stem ) and which are less important ( leaves ) and which really do not provide much information (this part of the value is ignored or truncated out) e.g. Fish lengths min = 25, max = 5 and the digits are X X. X Frequency Tables and Histograms Can t always list every possible value for quantitative variables or the datasets get too large. We wish to summarize the data in some way. So, we create groupings (intervals, bins, classes) and assign each observation to a grouping based on the value of its quantitative variable. ) How many groupings or intervals (classes)? number of observations c = 5 = as a rule of thumb but adjust as needed e.g. for n = 4, use 7 or 8 classes or bins n 5

5 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-5 2) How big is each class (bin)? a) Should be equal-sized b) Choose a starting value slightly below the min value in the dataset and an ending value slightly above the max value in the dataset ending value starting b) Size of each class = c adjusted as needed (see next step) value e.g. DDT as shown in the stem-and-leaf plot ranges from.3 to 2. ppm with n = 4 Use range from to 4 (instead of the actual min and max values) since it divides nicely with c = 7 3) Construct each class or grouping: FREQUENCY TABLE Grouping Absolute Frequency Relative Frequency /4 > /4 >4-6 /4 > /4 >8- /4 >-2 >2-4 /4 TOTAL 4.

6 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-6 Histogram (a graphical display of the frequency table) display either the absolute or relative frequency Distribution of DDT Quantiles.% maximum % quartile % median % quartile 3.25.% minimum.3 Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N 4

7 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3-7 The Histogram, i.e. the frequency distribution, or stem-andleaf plot plays an important role in statistical analysis. As a consequence we spend a lot of time and effort describing these distributions. The descriptions include: ) Shape of the distribution (skew, modality, symmetry, gaps, and outlying or other unusual data points) Std. Dev = 6.98 Mean = 7.2 N = CONCENTR

8 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS Std. Dev = 9.9 Mean = 52. N = X Special name for distributions which follow a symmetric, unimodal shape with equal sized tails and with a specific curve between the mode and the tails: NORMAL DISTRIBUTION

9 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS Std. Dev = 2.8 Mean = 7. N = TIME 2) Center of the Distribution (topic 4) 3) Spread of the Distribution (topic 5) B) Frequency Distributions for Populations Imagine you have the data for an entire population rather than just a sample from that population (census). N = population size >>> n = sample size. If we used the rule of thumb for number of bars needed we d get an extremely large number:

10 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3- n = 25 8 The tops of the bars approach a smooth line this is called the density curve of the population 6 4 Another Example N=4 n = n=25 N=

11 Topic (3) SUMMARIZING DATA - TABLES AND GRAPHICS 3- N=