Statistics

Experiment Title:

Elementary School Statistics for Elementary School Kids

Objective:

To learn about the statistical concepts of data types, central tendancy, variability, distribution shape, and descriptive statistics.

Background:

Statistics is the study of data and the information that we can get from data. Statistics primarily concerns the uses and collections of data, functions of data, tables, charts, and other graphics to meaningfully communicate information.

Data

Data are typically numbers or labels that represent a measurement of some type. Examples of numeric measurements (sometimes called numeric measurements, scale data, or ratio data) include length, weight, height, volume, temperature, time, voltage, current, and pressure. Measurements that might be identified as labels (sometimes called qualitative data, nominal data, or categorical data) include hair color, eye color, whether a coin lands on heads or tails, the outcome of a dice roll, whether a person likes chocolate, and a person’s favorite subject in school.

A way to tell if a variable is qualitative is whether numbers can be assigned to labels in any order without changing the meaning of of how the data are labeled.

For example, if we have 3 eye colors: blue = 1, brown = 2, green = 3, we could reassign blue = 4, brown = 2, and green = 6 without any meaningful difference to the numbers (because the numbers are assigned arbitrarily and differences in the numbers used for the labels have nothing to do with the labels themselves).

If we have 3 height measurements: 20 cm (labeled 1), 40 cm (labeled 2), and 60 cm (labeled 3), and we reassign the labels as 20 cm (labeled 2), 40 cm (labeled 1), and 60 cm (labeled 3), we would no longer have a relationship between the differences in the labels and the observed lengths.

Time Ordering

Some data is collected over time and the measurement of time is meaningful to the interpretation of data. Data that are collected over time are often called time series data. Data that represent a single point in time are called cross-sectional data.

An example of time series data are the time-ordered measurements of height over a person’s lifetime.
An example of cross sectional data are the list of heights of all first graders on a particular day of the year.

Time ordering can play a special role in the analysis of data and may provide insight into whether measurements change over time.

Functions of Data

Statistics uses functions of data to summarize a number of data points or relationships between data points.

Often two important questions in analyzing a set of data are “what does a typical data point look like?” and “how different are the data in the data set collected from the ‘typical’ value?” Using other labels and subgroups of the data, we can consider differences between groups. As an example, if we were studying height, we might be curious whether gender plays a role in height (i.e. are boys of a certain age taller or shorter than girls of the same age?). We might also be interested in whether geography plays a role (i.e. Are children in Colorado taller or shorter than children in Florida at the same age?).

For numeric data, one measure of the “typical” value is called the mean. Consider that we measure 4 ordinary objects and get the following measurements: 2 inches, 4 inches, 4 inches, and 6 inches. To find the mean we add the measurements together and divide by the number of measurements, i.e. 2+4+4+6 = 16. 16 / 4 = 4. The mean of our measurements is 4. Besides acting as a ‘balancing point’, the mean acts as an estimate of what we might get if we measure another item.
If we want to consider “how different are our measurements from the typical value,” we could subtract the mean from each measurement and take the absolute value:

• |2−4|=|−2|=2
• |4−4|=|0|=0
• |4−4|=|0|=0
• |6−4|=|2|=2

Taking the average of these deviations, (2+0+0+2)/4 = 1 indicates that on average, each data point might be 1 unit away from the “typical value” for our particular measurements. This value is called the “mean absolute deviation” and is a valid measure of the “spread” of the data.

Another measure of the “spread” of the data is called the standard deviation. It is calculated as the square root of the sum of squared differences from the mean divided by one less than the number of measurements (this is also called the sample standard deviation). For example, in our measurements:

(2−4)2+(4−4)2+(4−4)2+(6−4)24−1−−−−−−−−−−−−−−−−−−−√=83−−√=1.63

For many data sets that are fit by a bell-shaped curve, the standard deviation gives us an idea of where we might expect to find individual measurements in our data set and measurements that we have not measured yet that are similar to the data that we have collected. This is conceptually the probability of getting a specific measurement. The diagram below shows the relationship between mean, standard deviations from the mean, and probability of finding a point.

For categorical data, one measure of the “typical” value is called the mode. The mode is the label that occurs most often in a set of data. For example, in a set of coin tosses: H, H, T, T, T, T, H, H, T, T, we have 6 Tails and 4 Heads. Tails occurs most often and the mode of the data would be said to be Tails. The mode of the data tells us what we might expect to obtain if we measure another item.

Measures of dispersion for categorical data are less standardized. One commonly used measure for categorical dispersion is the difference between the category with the largest number of values and the category with the smallest number of values. For the coin toss example above, we had 6 Tails and 4 Heads. This sample would be said to have a range of 6−4=2.

An important concept that we will be using in the lab today is the concept of a quantile. Quantiles are meaningful for numeric measurements. Consider the population of all students in your own grade. This includes both your school and the rest of the world. Imagine that we line all students in your grade up from shortest to tallest. Are you generally shorter, taller, or in the middle? Are you at one of the ends of the distribution? We might consider an individual measurement in terms of its place in the larger distribution:

• Percentiles - Find measurement values that evenly place 1% of the population in each group (100 groups total).
• Quartiles - Find measurement values that evenly place 25% of the population in each group (4 groups total).
• Deciles - Find measurement values that evenly place 10% of the population in each group (10 groups total).
• Quintiles - Find measurement values that evenly place 20% of the population in each group (5 groups total).

There are some special quantile values that are often used in communicating information:

• Minimum - The smallest measurement
• First quartile - the measurement that separates the bottom 25% and top 75%
• Median - The measurement that separates the bottom 50% and top 50%
• Third quartile - The measurement that separates the bottom 75% and top 25%
• Maximum - The largest measurement

Taken together, the numbers above are called the “five number summary.”
Not all numeric populations correspond to a bell shaped distribution. Distributions that are not evenly distributed around the mean value are considered skewed. Below is an example of the Chi Square distribution, which is considered to have positive skewness (also called skewed right).

Today, we will primarily focus on two activities.

Materials

For the first activity the following materials will be needed:

• Roll of paper for measuring height
• Tape
• Pens/Markers
• Measuring tape or meter/yard sticks
• Growth Tables / Charts for males and females (source: Centers for Disease Control and Prevention, National Center for Health Statistics. “Growth Charts - Data Table of Stature-for-age Charts.” https://www.cdc.gov/growthcharts/html_charts/statage.htm. Accessed 4/6/2018)

For the second activity the following material will be needed:

• Bead Bags
• Container to mix bead bags

Procedure

Activity 1

1. Split students into groups by grade or by age
2. For all students in the group, identify ages to the nearest month
3. Using the paper, trace the outline of the students in a way that allows height to be measured. Consider marking heights along the paper from shortest to tallest. Note the student’s name, age, and gender. Identify the student’s height in inches
4. Using the tables and charts below, determine the approximate quantile for each student based on age, gender, and height. Write it with the outline of the student.
5. Determine the average height for each group, then compute the average height for all students

Data table for Females (measurement in inches):

Months	3rd	5th	10th	25th	50th	75th	90th	95th	97th
60.5	39.1	39.5	40.2	41.3	42.5	43.8	45.0	45.7	46.2
61.5	39.3	39.7	40.4	41.5	42.7	44.0	45.2	46.0	46.5
62.5	39.5	39.9	40.6	41.7	43.0	44.3	45.5	46.2	46.7
63.5	39.7	40.2	40.8	41.9	43.2	44.5	45.7	46.5	47.0
64.5	40.0	40.4	41.0	42.1	43.4	44.8	46.0	46.7	47.2
65.5	40.2	40.6	41.2	42.4	43.7	45.0	46.2	47.0	47.5
66.5	40.4	40.8	41.5	42.6	43.9	45.2	46.5	47.3	47.8
67.5	40.6	41.0	41.7	42.8	44.1	45.5	46.7	47.5	48.0
68.5	40.8	41.2	41.9	43.0	44.4	45.7	47.0	47.8	48.3
69.5	41.0	41.4	42.1	43.3	44.6	46.0	47.2	48.0	48.5
70.5	41.2	41.7	42.3	43.5	44.8	46.2	47.5	48.3	48.8
71.5	41.4	41.9	42.5	43.7	45.0	46.4	47.7	48.5	49.1
72.5	41.6	42.1	42.8	43.9	45.3	46.7	48.0	48.8	49.3
73.5	41.8	42.3	43.0	44.2	45.5	46.9	48.2	49.0	49.6
74.5	42.1	42.5	43.2	44.4	45.7	47.2	48.5	49.3	49.8
75.5	42.3	42.7	43.4	44.6	46.0	47.4	48.7	49.5	50.1
76.5	42.5	42.9	43.6	44.8	46.2	47.6	49.0	49.8	50.3
77.5	42.7	43.1	43.8	45.0	46.4	47.9	49.2	50.0	50.6
78.5	42.9	43.3	44.0	45.2	46.6	48.1	49.4	50.3	50.8
79.5	43.1	43.5	44.2	45.5	46.9	48.3	49.7	50.5	51.1
80.5	43.3	43.7	44.4	45.7	47.1	48.5	49.9	50.8	51.3
81.5	43.5	43.9	44.6	45.9	47.3	48.8	50.1	51.0	51.5
82.5	43.7	44.1	44.8	46.1	47.5	49.0	50.4	51.2	51.8
83.5	43.9	44.3	45.0	46.3	47.7	49.2	50.6	51.5	52.0
84.5	44.0	44.5	45.2	46.5	47.9	49.4	50.8	51.7	52.3
85.5	44.2	44.7	45.4	46.7	48.1	49.7	51.1	51.9	52.5
86.5	44.4	44.9	45.6	46.9	48.4	49.9	51.3	52.1	52.7
87.5	44.6	45.1	45.8	47.1	48.6	50.1	51.5	52.4	52.9
88.5	44.8	45.3	46.0	47.3	48.8	50.3	51.7	52.6	53.2
89.5	45.0	45.4	46.2	47.5	49.0	50.5	51.9	52.8	53.4
90.5	45.1	45.6	46.4	47.7	49.2	50.7	52.2	53.0	53.6
91.5	45.3	45.8	46.6	47.9	49.4	50.9	52.4	53.2	53.8
92.5	45.5	46.0	46.8	48.1	49.6	51.1	52.6	53.5	54.0
93.5	45.7	46.2	46.9	48.2	49.8	51.3	52.8	53.7	54.3
94.5	45.8	46.3	47.1	48.4	49.9	51.5	53.0	53.9	54.5
95.5	46.0	46.5	47.3	48.6	50.1	51.7	53.2	54.1	54.7
96.5	46.2	46.7	47.5	48.8	50.3	51.9	53.4	54.3	54.9
97.5	46.3	46.8	47.6	49.0	50.5	52.1	53.6	54.5	55.1
98.5	46.5	47.0	47.8	49.1	50.7	52.3	53.8	54.7	55.3
99.5	46.7	47.2	48.0	49.3	50.9	52.5	54.0	54.9	55.5
100.5	46.8	47.3	48.1	49.5	51.1	52.7	54.2	55.1	55.7
101.5	47.0	47.5	48.3	49.7	51.2	52.9	54.4	55.3	55.9
102.5	47.1	47.6	48.4	49.8	51.4	53.0	54.5	55.5	56.1
103.5	47.3	47.8	48.6	50.0	51.6	53.2	54.7	55.7	56.3
104.5	47.4	47.9	48.7	50.1	51.7	53.4	54.9	55.9	56.5
105.5	47.6	48.1	48.9	50.3	51.9	53.6	55.1	56.0	56.7
106.5	47.7	48.2	49.1	50.5	52.1	53.7	55.3	56.2	56.8
107.5	47.8	48.4	49.2	50.6	52.2	53.9	55.5	56.4	57.0
108.5	48.0	48.5	49.3	50.8	52.4	54.1	55.7	56.6	57.2
109.5	48.1	48.7	49.5	50.9	52.6	54.3	55.8	56.8	57.4
110.5	48.2	48.8	49.6	51.1	52.7	54.4	56.0	57.0	57.6
111.5	48.4	48.9	49.8	51.2	52.9	54.6	56.2	57.2	57.8
112.5	48.5	49.1	49.9	51.4	53.1	54.8	56.4	57.3	58.0
113.5	48.7	49.2	50.1	51.6	53.2	55.0	56.6	57.5	58.2
114.5	48.8	49.3	50.2	51.7	53.4	55.1	56.7	57.7	58.4
115.5	48.9	49.5	50.4	51.9	53.6	55.3	56.9	57.9	58.6
116.5	49.1	49.6	50.5	52.0	53.7	55.5	57.1	58.1	58.8
117.5	49.2	49.8	50.7	52.2	53.9	55.7	57.3	58.3	59.0
118.5	49.3	49.9	50.8	52.3	54.1	55.9	57.5	58.5	59.2
119.5	49.5	50.0	50.9	52.5	54.2	56.0	57.7	58.7	59.4
120.5	49.6	50.2	51.1	52.6	54.4	56.2	57.9	58.9	59.6
121.5	49.7	50.3	51.2	52.8	54.6	56.4	58.1	59.1	59.8
122.5	49.9	50.5	51.4	53.0	54.8	56.6	58.3	59.3	60.0
123.5	50.0	50.6	51.6	53.1	55.0	56.8	58.5	59.5	60.2
124.5	50.2	50.8	51.7	53.3	55.1	57.0	58.7	59.7	60.4
125.5	50.3	50.9	51.9	53.5	55.3	57.2	58.9	60.0	60.6
126.5	50.5	51.1	52.0	53.7	55.5	57.4	59.1	60.2	60.9
127.5	50.6	51.2	52.2	53.9	55.7	57.6	59.4	60.4	61.1
128.5	50.8	51.4	52.4	54.1	55.9	57.8	59.6	60.6	61.3
129.5	50.9	51.6	52.6	54.3	56.1	58.1	59.8	60.9	61.5
130.5	51.1	51.8	52.8	54.5	56.4	58.3	60.0	61.1	61.8
131.5	51.3	51.9	53.0	54.7	56.6	58.5	60.3	61.3	62.0
132.5	51.5	52.1	53.1	54.9	56.8	58.7	60.5	61.6	62.3
133.5	51.7	52.3	53.4	55.1	57.0	59.0	60.7	61.8	62.5
134.5	51.8	52.5	53.6	55.3	57.3	59.2	61.0	62.0	62.7
135.5	52.1	52.7	53.8	55.5	57.5	59.5	61.2	62.3	63.0
136.5	52.3	52.9	54.0	55.8	57.7	59.7	61.5	62.5	63.2
137.5	52.5	53.2	54.2	56.0	58.0	59.9	61.7	62.8	63.5
138.5	52.7	53.4	54.5	56.2	58.2	60.2	61.9	63.0	63.7
139.5	52.9	53.6	54.7	56.5	58.4	60.4	62.2	63.2	63.9
140.5	53.1	53.8	54.9	56.7	58.7	60.7	62.4	63.5	64.1
141.5	53.4	54.1	55.2	57.0	58.9	60.9	62.6	63.7	64.4
142.5	53.6	54.3	55.4	57.2	59.2	61.1	62.9	63.9	64.6
143.5	53.9	54.6	55.6	57.4	59.4	61.4	63.1	64.1	64.8
144.5	54.1	54.8	55.9	57.7	59.6	61.6	63.3	64.4	65.0
145.5	54.4	55.1	56.1	57.9	59.9	61.8	63.5	64.6	65.2
146.5	54.6	55.3	56.4	58.1	60.1	62.0	63.7	64.8	65.4
147.5	54.8	55.5	56.6	58.4	60.3	62.2	63.9	65.0	65.6
148.5	55.1	55.8	56.8	58.6	60.5	62.4	64.1	65.1	65.8
149.5	55.3	56.0	57.1	58.8	60.7	62.6	64.3	65.3	66.0

Growth Chart for Females (measurement in inches):

Data table for Males (measurement in inches):

Months	3rd	5th	10th	25th	50th	75th	90th	95th	97th
60.5	39.5	39.9	40.6	41.7	43.0	44.2	45.3	46.0	46.4
61.5	39.7	40.1	40.8	41.9	43.2	44.4	45.5	46.2	46.6
62.5	39.9	40.3	41.0	42.2	43.4	44.7	45.8	46.4	46.9
63.5	40.1	40.5	41.2	42.4	43.6	44.9	46.0	46.7	47.1
64.5	40.3	40.7	41.4	42.6	43.8	45.1	46.2	46.9	47.3
65.5	40.4	40.9	41.6	42.8	44.1	45.3	46.5	47.1	47.6
66.5	40.6	41.1	41.8	43.0	44.3	45.5	46.7	47.4	47.8
67.5	40.8	41.3	42.0	43.2	44.5	45.8	46.9	47.6	48.1
68.5	41.0	41.5	42.2	43.4	44.7	46.0	47.2	47.9	48.3
69.5	41.2	41.7	42.4	43.6	44.9	46.2	47.4	48.1	48.5
70.5	41.4	41.9	42.6	43.8	45.1	46.4	47.6	48.3	48.8
71.5	41.6	42.1	42.8	44.0	45.3	46.7	47.8	48.6	49.0
72.5	41.8	42.2	43.0	44.2	45.5	46.9	48.1	48.8	49.3
73.5	42.0	42.4	43.2	44.4	45.7	47.1	48.3	49.0	49.5
74.5	42.2	42.6	43.4	44.6	46.0	47.3	48.5	49.3	49.7
75.5	42.3	42.8	43.6	44.8	46.2	47.5	48.8	49.5	50.0
76.5	42.5	43.0	43.8	45.0	46.4	47.8	49.0	49.7	50.2
77.5	42.7	43.2	44.0	45.2	46.6	48.0	49.2	50.0	50.4
78.5	42.9	43.4	44.2	45.4	46.8	48.2	49.4	50.2	50.7
79.5	43.1	43.6	44.4	45.6	47.0	48.4	49.7	50.4	50.9
80.5	43.3	43.8	44.6	45.8	47.2	48.6	49.9	50.7	51.2
81.5	43.5	44.0	44.7	46.0	47.4	48.8	50.1	50.9	51.4
82.5	43.7	44.2	44.9	46.2	47.6	49.1	50.3	51.1	51.6
83.5	43.9	44.4	45.1	46.4	47.8	49.3	50.6	51.4	51.9
84.5	44.1	44.6	45.3	46.6	48.0	49.5	50.8	51.6	52.1
85.5	44.3	44.8	45.5	46.8	48.2	49.7	51.0	51.8	52.3
86.5	44.5	44.9	45.7	47.0	48.5	49.9	51.2	52.0	52.6
87.5	44.6	45.1	45.9	47.2	48.7	50.1	51.5	52.3	52.8
88.5	44.8	45.3	46.1	47.4	48.9	50.3	51.7	52.5	53.0
89.5	45.0	45.5	46.3	47.6	49.1	50.5	51.9	52.7	53.3
90.5	45.2	45.7	46.5	47.8	49.3	50.8	52.1	52.9	53.5
91.5	45.4	45.9	46.7	48.0	49.5	51.0	52.3	53.2	53.7
92.5	45.6	46.1	46.8	48.2	49.7	51.2	52.6	53.4	53.9
93.5	45.7	46.2	47.0	48.4	49.9	51.4	52.8	53.6	54.2
94.5	45.9	46.4	47.2	48.5	50.1	51.6	53.0	53.8	54.4
95.5	46.1	46.6	47.4	48.7	50.2	51.8	53.2	54.0	54.6
96.5	46.3	46.8	47.6	48.9	50.4	52.0	53.4	54.3	54.8
97.5	46.4	46.9	47.7	49.1	50.6	52.2	53.6	54.5	55.0
98.5	46.6	47.1	47.9	49.3	50.8	52.4	53.8	54.7	55.3
99.5	46.8	47.3	48.1	49.5	51.0	52.6	54.0	54.9	55.5
100.5	46.9	47.5	48.3	49.6	51.2	52.8	54.2	55.1	55.7
101.5	47.1	47.6	48.4	49.8	51.4	53.0	54.4	55.3	55.9
102.5	47.3	47.8	48.6	50.0	51.6	53.2	54.6	55.5	56.1
103.5	47.4	48.0	48.8	50.2	51.8	53.4	54.8	55.7	56.3
104.5	47.6	48.1	48.9	50.3	51.9	53.6	55.0	55.9	56.5
105.5	47.7	48.3	49.1	50.5	52.1	53.7	55.2	56.1	56.7
106.5	47.9	48.4	49.3	50.7	52.3	53.9	55.4	56.3	56.9
107.5	48.0	48.6	49.4	50.9	52.5	54.1	55.6	56.5	57.1
108.5	48.2	48.7	49.6	51.0	52.7	54.3	55.8	56.7	57.3
109.5	48.3	48.9	49.7	51.2	52.8	54.5	56.0	56.9	57.5
110.5	48.5	49.0	49.9	51.4	53.0	54.7	56.2	57.1	57.7
111.5	48.6	49.2	50.1	51.5	53.2	54.9	56.4	57.3	57.9
112.5	48.8	49.3	50.2	51.7	53.3	55.0	56.6	57.5	58.1
113.5	48.9	49.5	50.4	51.8	53.5	55.2	56.8	57.7	58.3
114.5	49.1	49.6	50.5	52.0	53.7	55.4	57.0	57.9	58.5
115.5	49.2	49.8	50.6	52.1	53.8	55.6	57.1	58.1	58.7
116.5	49.3	49.9	50.8	52.3	54.0	55.7	57.3	58.3	58.9
117.5	49.5	50.0	50.9	52.5	54.2	55.9	57.5	58.5	59.1
118.5	49.6	50.2	51.1	52.6	54.3	56.1	57.7	58.6	59.3
119.5	49.7	50.3	51.2	52.8	54.5	56.3	57.9	58.8	59.5
120.5	49.9	50.5	51.4	52.9	54.7	56.4	58.0	59.0	59.7
121.5	50.0	50.6	51.5	53.1	54.8	56.6	58.2	59.2	59.8
122.5	50.1	50.7	51.7	53.2	55.0	56.8	58.4	59.4	60.0
123.5	50.3	50.9	51.8	53.4	55.1	56.9	58.6	59.6	60.2
124.5	50.4	51.0	51.9	53.5	55.3	57.1	58.8	59.8	60.4
125.5	50.5	51.1	52.1	53.7	55.5	57.3	58.9	59.9	60.6
126.5	50.7	51.3	52.2	53.8	55.6	57.4	59.1	60.1	60.8
127.5	50.8	51.4	52.4	54.0	55.8	57.6	59.3	60.3	61.0
128.5	50.9	51.6	52.5	54.1	55.9	57.8	59.5	60.5	61.2
129.5	51.1	51.7	52.7	54.3	56.1	58.0	59.6	60.7	61.3
130.5	51.2	51.8	52.8	54.4	56.3	58.1	59.8	60.9	61.5
131.5	51.4	52.0	52.9	54.6	56.4	58.3	60.0	61.0	61.7
132.5	51.5	52.1	53.1	54.7	56.6	58.5	60.2	61.2	61.9
133.5	51.6	52.3	53.2	54.9	56.8	58.6	60.4	61.4	62.1
134.5	51.8	52.4	53.4	55.1	56.9	58.8	60.6	61.6	62.3
135.5	51.9	52.6	53.6	55.2	57.1	59.0	60.8	61.8	62.5
136.5	52.1	52.7	53.7	55.4	57.3	59.2	61.0	62.0	62.7
137.5	52.2	52.9	53.9	55.5	57.4	59.4	61.1	62.2	62.9
138.5	52.4	53.0	54.0	55.7	57.6	59.6	61.3	62.4	63.1
139.5	52.6	53.2	54.2	55.9	57.8	59.8	61.5	62.6	63.3
140.5	52.7	53.4	54.4	56.1	58.0	60.0	61.8	62.8	63.6
141.5	52.9	53.5	54.5	56.3	58.2	60.2	62.0	63.1	63.8
142.5	53.1	53.7	54.7	56.4	58.4	60.4	62.2	63.3	64.0
143.5	53.2	53.9	54.9	56.6	58.6	60.6	62.4	63.5	64.2
144.5	53.4	54.1	55.1	56.8	58.8	60.8	62.6	63.7	64.5
145.5	53.6	54.3	55.3	57.0	59.0	61.0	62.8	64.0	64.7
146.5	53.8	54.4	55.5	57.2	59.2	61.2	63.1	64.2	64.9
147.5	54.0	54.6	55.7	57.4	59.4	61.4	63.3	64.4	65.2
148.5	54.2	54.8	55.9	57.6	59.6	61.7	63.5	64.7	65.4
149.5	54.3	55.0	56.1	57.9	59.9	61.9	63.8	64.9	65.7

Growth Chart for Males (measurement in inches):

Activity 2

1. Identify the color of bead that needs to be counted.
2. Select a bag of beads from the container at random
3. Count the number of beads in the bag that match the target color and record the count in your lab notebook.
4. Put all of the beads back in the bag, close it securely, return it to the bin and mix the bin.
5. Repeat process 1-4 5-10 times.
6. Working with other students, combine measurements and come up with a count for each outcome (ex. 4 measurements had 1 white bead, 10 measurements had 2 white beads, etc).
7. Create a table and chart of your results.

Conclusions

Activity 1

1. Where was your height measurement in relation to the rest of the population that is close to your age? People run the full spectrum of height.
2. Based on your current approximate quantile, how tall might you be in 1 year if you stay at the current quantile?
3. Is your entire group similar in height? Is your group clustered in a particular area of the growth charts? On average we might expect most people to be between the 25th and 75th population percentiles, but you might have people (or with significantly less probability an entire group) that are at one extreme or the other based on age.

Activity 2

1. What is the shape of your graph? Does it appear to be symmetric or skewed? Depending on how the bags were set up, it is possible to get a number of results. Some could be symmetric, others might be left or right skewed. The parent/teacher instructions for the lab provide several different possibilities.
2. What was the purpose of returning the bag to the container and mixing the container after the bag was returned? If we simply counted each bag once, would we get the same appearance on the graph of the results?

Extension Activities

1. What would you currently think causes differences in height between people of the same age? Are these due to person-person differences, life history, environment, genetics, a combination of factors, or something else entirely?
2. Consider the growth charts used for Activity 1 above. Between younger ages and older ages, does the difference between 97th percentile and 3rd percentile stay the same, increase, or decrease? Why might this occur?
3. The growth charts and tables specifically do not provide minimum or maximum heights for any age and appear to leave out the most extreme points. Why might this be?
4. Consider two samples: 1. A random sample of 25 third graders and 2. A random sample of 25 students from grades 1-5. Would the average height of the two groups be close to the same or have a large difference between the samples? Would the dispersion (standard deviation or mean absolute deviation as it was introduced in this lab) between the two groups be close to the same or have a large difference? Why might this be?
5. Consider the chart below that shows the 3rd percentile, median (50th percentile), and 97th percentile for males (orange) and females (blue). How would you settle the debate around whether boys or girls are taller for the ages shown in the chart? If a student at another school said that the girls were taller than the boys in 3rd grade (roughly 90-108 months old), what would you say based on your knowledge of sampling and the population data in the charts in this lab?