AP Biology: Standard Deviation and Standard Error of the Mean Carol Leibl National Math and Science Initiative Dallas, TX

Experimental Science In experimental science, observations and measurements are made. Each measurement represents a piece of data. When multiple measurements are made or data collected, one might want to analyze the data to determine trends and conclusions.

Types of Data When measuring the amount of oxygen being consumed, that is continuous data. Counting the number of males versus females is an example of discrete data.

Averaging or Determining What is Normal Determining trends: Averaging or tendency toward a central value. Mean Median Mode

Experimental Science Background: Mean- The average of a set of numbers or the calculated central value. It the sum of the numbers divided by the number of values used n=the number of values used n Xi is the sum of all the values. i=1 The mean is equal to the sum of the numbers divided by the number of values used. Experimental Science

Averaging or Determining What is Normal The median is the middle number (in a sorted list of numbers). To find the median, place the numbers you are given in value order and find the middle number. Example: find the median of {13, 23, 11, 16, 15, 10, 26}. Put them in order: {10, 11, 13, 15, 16, 23, 26} and 15 is the median value.

Averaging or Determining What is Normal The mode is the value that appears most often (in a sorted list of numbers). 13, 13, 13, 13, 14, 14, 16, 18, 21 13 is the mode from the list above. The median is 14, while the mean is 15.

The daily high temperature for the last nine days has been recorded. Your Turn The daily high temperature for the last nine days has been recorded. Determine the mean, mode and median. Day Temperature oF 1 54 2 45 3 68 4 67 5 6 52 7 63 8 9

Answer Day Temperature oF 1 45 2 52 3 4 54 5 63 6 67 7 68 8 9 537 60 Median 6 67 7 68 8 Mode 9 SUM 537 Mean 537/9 60

Frequency Distribution Table and Histograms Frequency distribution table shows how often a measurement occurs in a collection of data. Histogram is a type of bar graph that illustrates a frequency distribution. The horizontal represents the continuous measurements. The vertical axis represents the frequency of the measurements.

Frequency Distribution Table and Histograms Quite often there are too many data points to make individual bars so the data is grouped together in increments that have a particular range. These increments are called bins.

Example A student wanted to measured the mass of 300 mung beans to the nearest 0.001 g

Example After measuring a sample, the student decided that the bin size should be in increments of .01 g. As each bean was massed, the mass was recorded in a frequency distribution table and put into container, representing a bin.

Creating a Histogram . Then a histogram was made illustrating the frequency distribution of the masses. Each bar represents 0.01 g

If a number of data points are plotted on a histogram and the majority of points lie close to the average and fewer data points are found further from the average, it forms a bell shaped curve. This is called the normal distribution. If it is a true normal distribution, then the mean, mode and median should all occur at the same place. Experimental Science

Statistics, Analysis, and AP Biology Many things when measured and plotted from a large population tend to form curves that approximate the normal distribution. These are World War I soldiers arranged in rows according to height. While the curve formed by these soldiers is not perfectly symmetrical, there is a cluster around the mean with rapidly tapering tails. Many traits like height, weight, and I.Q. form normal distributions. Statistics, Analysis, and AP Biology

Creating a Histogram . The data from the mung beans approximates a normal distribution 0.01 g.

Your Turn Obtain a container from your teacher with a sample of seeds and some containers to be used as bins. Mass 30 sample seeds Record the mass. Place them in the correct bin. Determine the mass. Make a histogram based on the number of seeds found in each bin. Does it form a normal distribution? You can have students combine containers to make a histogram with a larger sample size.

Histograms Using 300 Seeds

Two Different Normal Distributions How are these two histograms alike? How are they different? On the AP exam, both multiple choice and free response style questions visit this connection using graphical, analytical, and application-based stems. Let’s take a look at some recent free response questions that illustrate this concept:

Two Different Normal Distributions These two histograms have the same mean of five but they differ in their variation or range of values. Range=(Max value)- (Min value) Determine the range for the seeds that you measured the mass. Which has more variation pinto beans or black-eyed peas?

Histograms Using 300 Seeds

AP Biology Content Specialist Carol Leibl AP Biology Content Specialist cleibl@nms.org