MODULE 4
MATHEMATICS AS A TOOL FOR BUSINESS AND FINANCE
Lesson No.  2 
Lesson Title  Descriptive Measures 
Specific Learning Outcomes  During the learning engagements, students should be able to: 1. Relate basic concepts of averages and deviations to normal distribution; and 2. Compute for means and deviations, and areas under the normal distribution function 
PROCESSING
A powerful tool in statistics is the normal distribution. It allows us to make inferences about the
whole population based on some characteristics of a sample.
If we gather all the weights of 1,000 freshmen students in a given school, the values will tend to
crowd around the value that is representative of the weight of majority of the students. The
situation is shown by the hypothetical histogram below.
Mean, Median, and
Mode are closely located
here with each other…
Symmetry Line/
Median (Midpoint)
Belllike shape curve
Adding more and smaller weight ranges to the histogram makes the highest points of the bars
approximate a curve. This is the normal distribution curve, which is more commonly known as
the bell curve.
Check this out: Symmetry Line/ Median (Midpoint)
Mean or average is located around here
Higher frequency (mode) seen right of (or
greater than) the median…
Scores of around 1000 students in a 140item test
a. Is there a possibility that the curve is “not normal”? or the shape is not belllike
shape? –yes
b. Is there a possibility that the curve is not symmetric? –yes
c. If there is a possibility that the curve is not symmetric, then provide situations or
examples that support your idea:
 Mean, Median, and Mode (Central Tendency measures) are not likely similar or close to one another. Mostly of the scores are either above average or below average (depends where the skewness is directedif positively skewed or negatively skewed) Mode is higher/greater than the mean; or, Mean is higher/greater than the mode; Median is lower/lesser than the mean and mode; etc. 
 

d. What is the role of the measures of central tendency to the curve?
A measure of central tendency is a single value that attempts to describe a set of data by identifying the central
position within that set of data. As such, measures of central tendency are sometimes called measures of central
location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure
of central tendency that you are most familiar with, but there are others, such as the median and the mode.
The mean, median and mode are all valid measures of central tendency, but under different conditions, some
measures of central tendency become more appropriate to use than others. In the following sections, we will look at
the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate
to be used.
Intermission trivia: When to use Mean, Median, and Mode…?
There are times that you cannot use the measures of central tendency in some types (or levels) of
data or variable and there are only best measures of central tendency that must be used in
particular data/variable:
 Nominal data (lowest level of measurement) is defined as data that is used for naming or labeling variables, without any quantitative value. It is sometimes called “named” data – a meaning coined from the word nominal. 
o  Examples are: Gender, Sexes, Skin and Hair Color, Nationality, color of the eyes, 
etc.
Since these data are just counted but with no any quantitative value, MODE is the best central
tendency measure… (How many boys and girls are in this class? How many students are
having blue eyes? How many Paulinian students are naturally born as Filipinos? Etc)
 Ordinal data (2nd lower level of measurement) is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the 
categories is not known. These data exist on an ordinal scale, one of four levels of
measurement.. (in short, they are categorized or ranked…ordered, hence its name
“ordinal)
o Examples are: Honor students, Arranging of my Grade 10 subject grades from
lowest to highest, order of siblings in the family, etc.
Since these data are categorized and arranged orderly or accordingly, MEDIAN is the best
central tendency measure… The middle value or midpoint in the ranked data gives us the
CENTER point value of the group of data…
Interval data (higher level of measurement) is a type of data which is measured along a
scale, in which each point is placed at an equal distance (interval) from one another.
Interval data is one of the two types of discrete data (Interval and Ratio). An example of
interval data is the data collected on a thermometer—its gradation or markings are
equidistant.
Ratio Data (highest level of measurement) is defined as quantitative data, having the
same properties as interval data, with an equal and definitive ratio between each data
and absolute “zero” being treated as a point of origin. In other words, there can be no
negative numerical value in ratio data.
Hence, the difference between Interval and Ratio data is that only Ratio accepts the
ABSOLUTE VALUE OF ZERO~ “Zero literally means nothing for Ratio…” but for
Interval, Zero still has a value (example is the temperature… Zero degrees Celsius do not
mean that there is absolutely no temperature but it means the freezing point of matter)
Since these data are exact and quantitatively expressed, MEAN and MEDIAN are the best
central tendency measures…
In Statistical distribution, you cannot use Median for skewed distribution because the scores
are not concentrated in the middle point (see previous examples in the processing
portion)…Median is only best if the curve is NORMAL (bellshape) since the concentrated
scores are in the middle point.
THE zscore
The zscore measures how many standard deviations an observed score is from the mean. To
compute for the zscore, we use:
̅
̅
Please use the URL to watch videos on solving zscore, or simply scan the QR codes using your
smartphones to follow the link:
ck12.org normal distribution problems: zscore
By Khan Academy
https://www.youtube.com/watch?v=Wp2nVIzBsE8
Activity 1: Computing for Z
Consider ̅ and , get the z…
38 
34 
30 
26 
24 
a. What is the meaning of z in relationship with the mean and standard deviation? And vice
versa?
b. What happens when any value x is equal to the mean?
c. What happens when z is equal to zero?
Activity 2: Z and Normal Curve
The following table translates zscores to their corresponding areas under the normal curve:
zscore  Area  zscore  Area 
0.0  0.0000  1.6  0.4452 
0.2  0.0793  1.8  0.4641 
0.4  0.1554  2.0  0.4772 
0.6  0.2257  2.2  0.4861 
0.8  0.2881  2.4  0.4918 
1.0  0.3413  2.6  0.4953 
1.2  0.3849  2.8  0.4974 
1.4  0.4192  3.0  0.4987 
a. What is the role of zscore in relationship to the area under the normal curve?
b. How can you define the area under the normal curve?
c. What happens to the area when the zscore is negative?
Search online for a ztable to compute for the area of following zscores:
zscore  Area 
1.64  
2.89  
1.75  
0.01  
0.49  
0.33  
0.19  
2.55  
1.01  
1.11 
Please use the URL to watch videos on how to use ztable or normal distribution table, or simply
scan the QR codes using your smartphones to follow the link:
Normal Distribution Table – Ztable Introduction
By Jalayer Academy
https://www.youtube.com/watch?v=lgwT6tDniko&t=4s
Learn how to create a normal distribution curve given mean
and standard deviation
By Brian McLogan
https://www.youtube.com/watch?v=qEGYkkif6xU
Assessment
1. The average number of workbook copies sold at a stand is 80 and a standard deviation of
4. If the number of workbooks sold over a month follows a normal distribution,
determine the probability that 25 copies are sold
2. The following are the weights (in kgs) of students in a Math class of 40 students:
43  45  35  43 
38  42  44  37 
40  42  38  35 
45  39  38  36 
38  45  41  37 
40  42  35  41 
38  37  41  45 
40  44  41  44 
40  35  41  35 
45  40  40  39 
Construct a normal distribution curve and then solve for:
a. Mean and Standard Deviation
b. What is the probability that the person to be called has a weight of 42 kgs?
Sources
Mellosantos, Luiz Allan, et at. (2016). Math Connections in the Digital Age: Statistics and
Probability. Sibs Publishing House, Inc. Quezon City.
YouTube Presentation and Tutorials
ck12.org normal distribution problems: zscore by Khan Academy
Descriptive Statistics in Excel by Elliott Jardin Ph.D.
Excel Formulas: NORM.DIST (NORMDIST) by Brian Henry
Excel Random Number Generator by Barb Henderson
How to Calculate Mean and Standard Deviation in Excel? by Eugene O’Loughlin
Learn how to create a normal distribution curve given mean and standard deviation by
Brian McLogan
Normal Distribution – Explained Simply (part 1) by how2stats
Normal Distribution Table – Ztable Introduction by Jalayer Academy
Properties of a Normal Distribution by Steve Mays
Standard Deviation – Explained and Visualized by Jeremy Jones
Statistics Fundamentals: The Mean, Variance and Standard Deviation by StatQuest with
Josh Starmer
StatQuest: The Normal Distribution, Clearly Explained!!! by StatQuest with Josh Starmer
Using NORM.DIST by Jeff Davis
Developed by: JOENAMER P. OÑEZ (ACLC College of Butuan)
MARY ANN M. GOZON (Butuan Doctors’ College)