Basic Statistical Terms and Definitions

Transcription

1 I. Basics Basic Statistical Terms and Definitions Statistics is a collection of methods for planning experiments, and obtaining data. The data is then organized and summarized so that professionals can analyze the results and draw conclusions based on this data. Two Types of Statistics Descriptive statistics are numbers that are used to consolidate a large amount of information. Any average is a descriptive statistic. Batting averages, average daily rainfall, or average daily temperature are good examples of descriptive statistics. Descriptive statistics uses numerical and/or visual techniques to summarize or describe the data in a clear and effective manner. Inferential statistics are used when we want to draw conclusions. It is the process of using sample information to draw inferences or conclusions about a population. For example when we want to determine if some treatment is better than another, or if there are differences in how two groups perform. Data Data are observations that have been collected. The data must be collected by a process of random selection. If the data is collected in a biased fashion, then the results of the experiment will be biased or meaningless. Techniques for collecting data can be read about in chapter 1 of your textbook. A population is the complete collection of all elements to be studied. A census is the collection of data from every element in the population. A sample is a sub-collection of data from elements in a population. Other Notes on Data A parameter is a numerical measurement describing a characteristic of a population. A statistic is a numerical measurement describing a characteristic of a sample. Quantitative data consists of numbers representing counts. separated into categories by a non-numeric characteristic. Qualitative data are Discrete data result when the possible values of the data are finite (countable). Continuous data result when the possible values of the data are infinite. Some examples of continuous data types are temperature, velocity, liquid measure, time, volume and distance.

2 II. Organizing and Presenting Data 5 Important Characteristics of Data: Data Organizers 1. Distribution: The shape the data makes when placed on a graph. 2. Center: Data sets have important characteristics that fall at or near center. 3. Variability: The spread of your data set. 4. Outlier: Values that fall far away from the majority of the data values. An outlier lies in the tail of a distribution. 5. Time: Data often changes over time we will learn to study data and make predictions on how data may or may not change. A Stem & Leaf Plot is a data organizer that shows the shape (distribution) the data makes when graphed along with the actual data values that are involved in the data set. A back-to-back Stem & Leaf Plot are two plots sharing the same set of stems. The plots go in opposite directions, and are used to compare and contrast two different data sets; usually pre & post some experiment.

3 Frequency Distributions & Graphs A frequency distribution table is a table in which a data set has been divided into distinct groups, called classes, along with the number of data values that fall into each class, called the frequency. The class width is defined as the difference between two successive lower (or upper) class limits. The Formula for the approximate class width is: Range CW, this result will be rounded up to the next whole number. # classes Note: The range is defined as the highest data value subtracted by the lowest data value. Also, the formula for the class width is not set in stone. The class boundary of a class is defined as the midpoint of the upper class limit of one class and the lower class limit of the next class. The class mark is the midpoint of a class. It is used as the representative value from that class. Its symbol is xm. Some Guidelines for Creating Classes: 1. If your data has more than 20 values, you should create a grouped frequency distribution by grouping data values together into classes. 2. There should be approximately 5 to 15 mutually exclusive classes. "Mutually exclusive" means that a score can belong to only one class. Two non-mutually exclusive classes would be and since the scores 47, 48, and 49 could belong to either class. In a grouped frequency distribution, the classes must be mutually exclusive. 3. Do not omit any classes. All possible scores between the largest score and the smallest score of the data set must be included in the grouped frequency distribution. Even if a class has a frequency of zero, it is to be included in the list of classes. 4. The class width should be equal for all classes. If the class width were not equal for all classes, then we could not perform the statistical computations that use grouped frequency distributions.

4 Distribution & Shape There are many different-shaped frequency distributions. The three most popular are the bell-shaped & skewed distributions. Note that skewness is determined by the tail of the distribution. Some other distribution shapes are given below in Figure popular as the ones above, but serve their purpose. These are not as Graphs A histogram is a bar graph that represents a continuous variable. To show the continuity of the data, the bars are connected to one another with no gap or breaks between any two adjacent bars. It order to accomplish this, we use the class boundaries as the value connecting two adjacent bars.

5 Frequency Packet 1: Descriptive Statistics & Frequency Distributions Example The following frequency table & frequency histogram gives the glucose level of 180 randomly selected individuals. The table was created using 9 classes and the calculated class width is 19(rounded). Notice that the class boundaries are used as the label on the x-axis if the frequency histogram. CW Range # classes 9 Frequency Histogram of Glucose Levels Glucoselevel The following table is the frequency distribution table that was used to create the histogram above. Class Frequency Class Boundaries

6 Measures of Center: 1. Mode: The most frequently occurring value in a dataset. The mode is easy to calculate, but is of limited use. The strength of the mode is that it is easy to determine. However, its weakness is that there may be more than one mode in a set of data. If we want a single average for a distribution, then this promotes a problem. 2. Median: The middle value in a dataset, i.e. half the data have values greater than the median and the other half have values which are less. The median is less sensitive to outliers than the mean and thus a better measure than the mean for highly skewed distributions. 3. Mean: The mean is a good measure of the central tendency of a distribution. It represents an equal distribution of your data amongst all groups. The problem with the mean is it is unreliable when the distribution is skewed. We should note that any of the above measures of center can be an average or typical data value within a set of data. The measure of center used to describe the average value depends on the kind of data you are working with. Location of Center in Distribution For a symmetrical distribution, the mean, median and mode are all equal and located at the center of the distribution. This is not true for skewed data. The mean is not a resistant measure and gets pulled towards the outlier as shown in the following diagram: Skewed Right Distribution Skewed Left Distribution

7 Measures of Variability: Variability measures the amount of spread in a data set. The wider a graph looks, the higher the amount of variability the data set is said to have. 1. Range: The difference between the largest and the smallest value in the dataset. Since the range only takes into account two values from the entire dataset, it may be heavily influenced by outliers in the data. 2. Variance: The variance is computed as the average squared deviation of each number from its mean. The units on the variance are squared units. 3. Standard Deviation: The standard deviation is the square root of the variance. It is a measure of the consistency of your data set. The smaller the value of the standard deviation is, the more consistent your data set will be. The units on the standard deviation will be the same as the units of your raw data. 4. Coefficient of Variation: A statistical measure of the dispersion of data points in a data series around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean, and is a useful statistic to compare the degree of variation from one data series to another, even if the units of measure or magnitude of the data are different from one another. It is calculated using the following formula: CV

8 Sample Problems 1. Given below are ages of the US Presidents when they were inaugurated a. Construct a frequency table & frequency histogram of the data. Use 6 classes. b. What is the shape of the distribution? c. Find the mean, median, mode, range, variance, coefficient of variation and standard deviation. d. Use the empirical rule to determine the percent of data that falls within 1,2 & 3 standard deviations from the mean. Does the empirical rule back up your answer to part B? e. Are there any outliers? Justify your answer. f. Find the 5-number summary, and construct a box-whisker plot. 2. Math test scores for two students are given below: Student A: 50,70,70,70,70,70,90 Student B: 50,55,62,70,78,85,90 a. Compute the mean, median and mode for each student. b. Is the mean a good indication of a typical grade for each student? Explain. c. Calculate the range, variance and standard deviation of each student. Which student s scores are more consistent and why? d. If you took the grades of student A and added 5 points to each grade, describe what would happen to the mean & the standard deviation. e. Assume that student A & B were in different classes with different instructors. Would it still be appropriate to use standard deviation to measure consistency? Explain. 3. The following represent temperatures in the month of August here at NCC last year a. Construct a frequency table & frequency histogram of the data. Use 7 classes. b. What is the shape of the distribution? c. Find the mean, median, mode, range, variance, coefficient of variation and standard deviation. d. Use the empirical rule to determine the percent of data that falls within 1,2 & 3 standard deviations from the mean. Does the empirical rule back up your answer to part B? e. Are there any outliers? Justify your answer. f. Find the 5-number summary, and construct a box-whisker plot.

9 4. Baseball Stats a) Hank Aaron was an outfielder for the Braves from 1954 to Here is the number of home runs hit each year by Aaron: 13, 27, 26, 44, 30, 39, 40, 34, 45, 44, 24, 32, 44, 39, 29, 44, 38, 47, 34, 40, 20. Mark McGwire has played in the major leagues since 1986 as a first baseman for the Oakland A's and the St. Louis Cardinals. The number of home runs hit by McGwire in each year are: 3, 49, 32, 33, 39, 22, 42, 9, 9, 39, 52, 34, 24, 58, 70, 65, 32. Give mean, median, mode, range, variance, coefficient of variation & standard deviation for each player. Aaron: mean: median: mode: range: variance: coefficient of variation: standard deviation: McGwire: mean: median: mode: range: variance: coefficient of variation: standard deviation: b) What conclusion would you draw from a comparison of these data? Judging only from the information you calculated above, who would you say was the most consistent HR hitter? Explain your answer.