Math Review for Physiology Students (LPC)
A) Understanding Range
If a set of numerical readings are taken by a scientist or clinician, it is likely that not all of the readings will be identical. For example, a person’s cholesterol level will tend to vary according to the time of the day or depending on what foods have been recently consumed. One isolated measurement may not be enough to figure out the individual’s typical, day-to-day serum cholesterol level. Some measurements will be higher than others. Thus, often, several measurements of the same thing are more revealing than a single one. However, in the end you will want to give a single measurement that is representative of all the others. This can be the range, and the average or mean.
If all the values are arranged in numerical order, from lowest to highest or vice versa, the range of values is the difference between the highest and lowest values in the series. For example, suppose that cholesterol readings over several days are 120 mg/100mL, 205 mg/100mL, 176 mg/100mL, 147 mg/100mL, and 234 mg/100mL. The lowest value is 120, the highest is 234. The difference between these two measurements is 114, meaning that the person’s serum cholesterol varied by 114 mg/100mL over the days when the measurements were taken. A second, more informational way of expressing range is to state the extremes. In the example from above, we can say the patient’s cholesterol ranged between 120 - 234 mg/100mL.
1) Suppose that over a particularly rainy winter week the rainfall amount was measured at 2.5 cm on Sunday, 3.53 cm on Monday, 1.51 cm on Wednesday, 0.75 cm on Thursday, and 2.12 cm on Saturday. State the range of rainfall, which fell during this week in two different ways.
1) The range of rainfall varied by ________________
2) The rainfall ranged between ___________________
2) Suppose that a person interested in his daily urine production measured his urine over a five-day period at 1,940 mL, 796 mL, 2,121 mL, 2,051 mL and 1,996 mL; all on different days. State the range of urine volume over the course of the study, in two different ways.
1) The range of urine production varied by ________________
2) The urine production ranged between ___________________
B) Understanding Average and Mean
Another way to express data is to calculate the average. The average, or mean, is found by adding the values together and dividing by the total number of values. In the serum cholesterol example given above the sum of the measurements equals 882 mg/100mL. The total number of measurements taken was 5. Therefore, the average is 882 ÷ 5, or 176.4 mg/100mL. (By the way, the unit of measurement “mg/100 ml” is frequently used in laboratory reports, including blood [glucose].)
1) Calculate the average, or mean, of the rainfall per day during the week described above.
_________________
2) Apparently it did not rain on Tuesday and Friday, so the rainfall for those days was 0 cm. If these days were included in the average, would it increase or decrease the result? If you did include them, your result shows the average daily rain. If you didn’t include them, your result shows the average rain when it rains. The two averages have slightly different meanings. The lesson here is that it is important to be clear on what exactly an average is means.
3) What is the average or mean amount of urine produced per day in the example on page 1?
___________________mL
4) Suppose you need to calculate how much urine an outpatient’s kidneys produce each day. You tell the patient to measure his urine production over a week’s time. He took measurements every day, but was traveling on one of the days and therefore unable to measure his urine that day. Would it be accurate to add 0 mL into the total and divide by 7? Why or why not?
5) Go back and look at the urine volume values given on the previous page. Notice that one day’s measurement of urine is considerably lower than that of the other days, and therefore it lowered the average considerably. You might be justified in leaving that measurement out of your calculation if you knew it was due to extremely abnormal circumstances (i.e. the patient spilled the urine sample or didn’t drink any liquids that day) but otherwise you should include it.
C) Understanding Percent
Calculating Percent from Data
Percent (designated by the sign %) refers to the number of instances out of every hundred. Thus, if 25% of the students in an early morning physiology class have not eaten any breakfast, it means that 25 out of 100 students have not eaten breakfast.
However, calculating percent is usually not that easy, because the total number of instances being studied does not add up to a convenient 100. For example, assume that the above class consists of 20 students. If 25% have not eaten breakfast, exactly how many students is that? _________
What if the class was made up of 16 students and four of them have not eaten breakfast? This means __________% have not eaten breakfast.
This was easy to calculate mentally. But what if it is not that easy? Not to worry: there is an easy way to calculate percent, regardless of the total number in the sample. You only need to identify which is the part under consideration, and which is the whole. Then you divide the part by the whole, multiply by 100, and add the % sign. In the above example the whole is the 16 students in the class, and the part being studied is the four who have not eaten breakfast. Doing the calculation, 4 ¸ 16 = 0.25, and 0.25 x 100 = 25. Add the % sign and you have the result: Four students are 25% of the class.
Notice how easy it is to multiply mentally by 100. You just have to move the decimal point two places to the right. For example: 0.25 x 100 = 25, and 4 x 100 = 400. If no decimal point is visible, it is assumed to be at the end of the number, and zeros are added to fill any empty spaces, as in the above example.
1) Suppose that 5% of the students in the class have not yet bought their textbook. If there are 100 students in the class, how many have not yet bought their book? __________
2) Suppose that the class is made up of 40 students, and two of them have not yet bought their textbook. What % of them has not bought their book? ______________%
Be careful not to make the common mistake of always dividing the small number into the larger! In most percent problems, the part is smaller that the whole. However, it is possible to encounter a situation in which the part is greater than the whole. This is possible when 100% is considered the normal expected amount, and the number of instances being studied is greater than that.
For example, the normal annual rainfall for a particular year is 28 cm, but in a certain year the region gets 34.4 cm. Once again the rule to follow is to divide the part by the whole and multiply by 100, then add the % sign. In the above example the whole is the normal 28 cm and the part being studied is the 34.4 cm in that particular year. Doing the calculation 34.4 ¸ 28 = 1.22, and 1.22 x 100 = 122. The rainfall in that year, then was 122% of normal.
3) A patient’s blood cell count shows 15,000 white blood cells (WBCs) per µL of blood. A normal WBC count is about 9,000 cells/µL (the normal range is actually 4,000 to 11,000 WBCs/µL. What percent of normal is this particular patent’s WBC count?
_____________
4) Obesity is defined medically as 20% above a person’s ideal weight. If a person’s ideal weight is 72 kg, when (above what weight) would that person be considered obese? ________________kg
5) If a man’s ideal weight is 165 lbs. and he weighs 185 lbs., how many pounds above his ideal weight is he? What percent above his ideal weight is he?
______________lbs. above ideal weight
______________% above ideal weight
Calculating Data from Percent:
It is even easier to calculate the number of measurements or occurrences when you are given the percent. Remember that “percent” means “the number of instances out of one hundred”. A helpful way to understand percent is to translate “per” as “divided by”, and “-cent” as 100. Thus, 50% is the same as 50 ¸ 100, or 0.5.
It is easy to determine percent if the number of instances is exactly 100. Suppose that 100 victims of smoke inhalation are brought into a hospital. 92% have been sent home after basic treatment. It is easy to calculate mentally that 92 victims have been sent home and 8 have been retained for additional treatment.
It is really not much more difficult if the number of instances is not exactly 100. Suppose the number of smoke inhalation victims equals 133, and 92% needed only basic treatment. 92% = 0.92, and 0.92 x 133 = 122.36 victims (or realistically 122 victims).
6) In one year, 120 individuals were diagnosed with AIDS in a particular city. Three years later, 68% of those individuals had shown some improvement, while 3% had died. How many patients had shown improvement, and how many had died.
_________ showed improvement
__________died
D) Scientific or Exponential Notation
Scientists often need to work with numbers that range from the astronomical to the subatomic. Such numbers are cumbersome to work with. For example an important concept in chemistry is the “mole”, which is 602,000,000,000,000,000,000,000 atoms, molecules or particles of any sort. How much more convenient it is to write that number as 6.02 x1023 instead!
In biology extremely large or small numbers are frequently encountered. Bacteria range in size between 0.0002 – 0.005 mm (millimeters). The average number of red blood cells in an adult male is about 5,500,000 cells/µL. Ultramicroscopic features in cells are often measured in Angstrom (Å) units, an Å being 0.0000000001 m or 0.0000001 mm. These numbers can be written in a shorter, more manageable form using standard exponential (or scientific) notation.
For example, 100 can be written as “10 squared” or 102; 1,000 can be written as 103; 10,000 as 104 etc. Likewise, a number like 186,000 can be written as 1.86 x 105.
In the scientific notation system any number can be expressed as the product of a number between 1 and 10 (including 1 but not 10), and a power of 10. Thus, the last example would be mathematically correct but scientifically unconventional to write as 18.6 x 104.
Similarly, very small numbers can be expressed as the product of a number between 1 and 10 (but not including 10), and a negative power of 10. Thus, 0.1 can be expressed
10-1, 0.01 as 10-2 etc. One Å can be expressed as 10-10 m. Again the power is equal to the number of spaces you need to move the decimal point, but this time to the left, in order to restore the number to its original form. The power is given a negative sign.
If bacteria range in size between 0.0002 – 0.005 mm, we can also express this range as
2 x 10-4 to 5 x 10-3 mm.
1) Suppose that the national budget surplus is 3.3 trillion dollars ($ 3,300,000,000,000).
Express this number in standard scientific notation. _____________
2) Express the following numbers in scientific notation:
The diameter of a human red blood cell is about 0.0075 mm. _______________
The smallest viruses are about 0.000018 mm wide. _______________
3) Express the following numbers in conventional form:
The length of an Argentine ant is 2.4 x 10-1 cm. _______________
An electron weighs 9.11 x 10-28 g. ___________________________________
E) UNITS OF MEASUREMENT: THE METRIC SYSTEM
For a rich resource of metric information go to: http://lamar.colostate.edu/~hillger/
Introduction:
The Metric system (from the Greek word metrikos, meaning “measure”), which was first developed in late eighteenth century France, is worldwide employed as the standard in scientific literature and in the field of medicine. The modern definitions of the units used in the metric system are those adopted by the General Conference on Weights and Measures, which in 1960 established the International System of Units (also called the modern metric system) abbreviated SI in all languages.
Technically, the United States does use the metric system: It’s been the nation’s official system since 1893. Unfortunately, most Americans prefer to ignore the official system in favor of weights and measures that are relics of when North America was a British colony. In terms of the family of nations, America’s only kin on the metric issue are Liberia and Burma!!
There has been some progress in the move to metric. Think of 35 millimeter film or soft drinks sold in 1- and 2-liter bottles. Furthermore, anyone who reads nutrition labels encounters milligrams. Cars and other products seeking to compete overseas generally are made to metric specifications. In the hospital, drugs are prescribed in mg or g per kg body weight. Clinical laboratories will report routine blood profiles with listings such as
HDL Cholesterol: 53 mg/dL
Glucose: 104 mg/dL
Albumin: 4.9 g/dL
This means, that you will now need to get used to the metric terminology. In our class we will exclusively use the metric system. You will need to recall the definitions for the metric units of length, mass, volume, and temperature. You probably should have a good understanding of converting from one system to the other, but for the purpose of this course you will only need to convert from one unit to another within the metric system.
meter (m) – unit of length equal to 1,650,763.73 wavelengths in a vacuum of the orange-red line of the spectrum of krypton-86. (1 m = 1.09 yards)