Is the problemmigomendoza.weebly.com/uploads/5/4/7/4/54745209/... · Honeygirl Pulot-pukyutan feel hungry and have soup for lunch. She choose one of the 400 grams cans bought in the

Question # 1:

Is the problem

clearly stated?

Question # 2:

Does the problem

have a theoretical

rationale?

Question # 3:

How significant

is the problem?

Question # 4:

Is there a review of

the literature? If so, is

it relevant?

Question # 5:

How clearly are

the hypotheses

stated?

Question # 6:

Are operational

definitions

provided?

Question # 7:

Is the procedure (or

methodology) used to answer

the problem fully and

completely described? Was a

sample used?

Question # 8:Are there any probable

sources of error that might

influence the results of the

study? If so, have they been

controlled?

Question # 9:

Were statistical

techniques used to

analyze the data? If so,

were they appropriate?

Question # 10:

How clearly are

the results

presented?

Question # 11:Are the conclusions

presented clearly? Do the

data support the conclusions?

Did the researcher

overgeneralize his findings?

Question # 12:

What are the limitations

of the study? Are they

stated?

Note 1:

Learn to abbreviate (i.e. History

of Mathematics to HOM, United

States to US) to avoid the

monotonous tone in your critique

paper.

Note 2:

Being verbose is not

necessary in writing a

critique paper/academic

paper.

Note 3:

Observe consistency

in tenses of verb and

usage of words.

Note 4:

Detach critique

paper to a feature

article.

Note 5:

Select the best term/word

that fits to the idea you

would like to convey.

Note 6:

Be exact in terms/words so

your readers will not

become confused.

Note 7:

It is a formal paper,

so please do not use

slang words.

Note 8:

It is a formal paper,

shortening of word is

not allowed.

Note 8:Let your presentation be well reasoned

and objective. If you passionately

disagree (or agree) with the author, let

your passion inspire you to new heights

of thorough research and reasoned

argument.

Challenge your ideas...

“to

summarize”

Chapter 3SUMMARY STATISTICS

Summary Statistics

The purpose of summary statistics is to

replace a huge indigestible mass of

numbers (the data) by just one or two

numbers that, together, convey most

of the essential information.

Note:Different statistics emphasize

different aspects of the data

and it will not always be

evident which aspect is more

important.

Example:In the house of Parliament (United

Kingdom), the members of

Parliament (MPs) were debating the

need for road signs in Wales to give

directions in both Welsh and English.

MP A:Since less than 10% of the

population of Wales speak

Welsh it is unnecessary to

include directions in Welsh.

Reflect:

What can you

say? Do you agree

or disagree?

MP B:Over 90% of the area of Wales

is inhabited by a population

whose principal language is

Welsh – directions in Welsh are

essential.

Reflect:

What can you

say? Do you agree

or disagree?

Clarification:

Both are correct,

however they led to

opposite inferences.

Reminder:We have to be careful to

choose our summary

statistics to be

appropriate.

Summary Statistics for

Univariate Data

Measures of Location/Measures

of Central Tendency

Measures of

Dispersion/Measures of Spread

Measures of Position

Answer the question:

“What sort of size values

are we talking about?”

Measures of Spread

Answer the question:

“How much do the

values vary?”

Measures of

Location/Central Tendency

These are statistics that

summarize a distribution of

scores by reporting the most

typical or representative value

of the distribution.

Measures of Location/Measures

of Central Tendency

The Mode

The Media

The Mean

The Mode

The mode of a set of discrete

data is the single value that

occurs most frequently.

It has limited use.

Kinds of Mode

Unimodal

Bimodal

Multimodal

Unimodal

A unique outcome

that occur most

frequent.

Bimodal

The data is described as being

bimodal if there are two such

outcomes that occur with equal

frequency then there is no unique

mode.

Multimodal

The data is multimodal if

there are three or more

such outcomes that occur

most frequent.

Something to think

about…

Can there be an

absence of mode in

the set of data?

Example 3.1

At the supermarket Ms. Honeygirl Pulot-

pukyutan buy 8 cans of soup. According

to the information on the tins, four have

mass 400 grams, three have mass 425

grams and one has mass 435 grams. Find

the mode.

Example 3.2After unpacking the shopping, Ms.

Honeygirl Pulot-pukyutan feel hungry and

have soup for lunch. She choose one of

the 400 grams cans bought in the

previous example. What is an appropriate

description of the frequency distribution

of the remaining 7 masses?

The Median

It is the positional value. It

is the “midpoint” of the

distribution when data are

ranked according to size.

How do we find for the

median?

After all the observations have been

collected, they can be arranged in a

row in order of magnitude, with the

smallest on the left and the largest on

the right (or vice versa).

Example 3.4

Suppose the observed

values are 13, 34, 19, 22

and 16. Find the media.

For odd number of

observations

If n is odd and equal to

(2k+1), say, then the

median is the (k+1)th

ordered value.

Example 3.5

For the soup cans in example 3.1, the

values were:

four 400 grams, three 425 grams and one

435 grams

Find the median.

For even number of

observations

If n is even and equal to

2k, say, then the median is

the average of the kth and

the (k+1)th ordered values.

Example 3.6 A chemistry professor has an accurate weighing machine and two

children, Kiko and Kika, who are keen on playing conkers. One day, Kiko

and Kika collect some new conkers. On their return home, following a

dispute over who has the best conkers, they use their father’s balance to

determine the weights of the conkers (in grams). Their results are as follows:

Kiko 31.4 44.4 39.5 58.7 63.6 51.5 60.0

Kika 60.1 34.7 42.8 38.6 51.6 55.1 47.0 59.2

Which among the collection of conkers has the higher median

weight?

The Mean

This is the average of the

set of data. It is the center

of the gravity of the

distribution.

How to compute for the

Mean?

This is equal to the sum of

all the observed values

divided by the total

number of observations.

Formula:

𝒙 =𝒔𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒕𝒉𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆𝒔

𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔

In Algebra:

𝑥 =(𝑥1+ 𝑥2+⋯+ 𝑥𝑛)

𝑛

Something to think about…

What can you say about the

value of the mean to the value

of the mode in relation to

individual observed values?

Advantages of Measures of


If a mode exists it is

certain to have a value

that was actually

observed.

Advantages of Measures of


The median can be

calculated in some

cases where the mean

or mode cannot.

Disadvantages of Measures of


The mode may not be

unique (because two or

more values may be

equally frequent)



The mean may be

significantly affected by the

inclusion of a mistaken

observation or of an usual

observation.



The statistical properties

of the mode and the

median are difficult to

determine.



In practice much more

use is made of the mean

than of either of the other

two measures of location.

Time for you to Investigate:

1. How many four-legged pets does the typical family have?

(a)Use a tally chart to record the combined number of dogs, cats,

hamsters, etc. for each member of your class.

(b)Determine the mean, median, and mode of these values. Which

was easiest to calculate?

2. An organization wishes to estimate the total number of four-

legged pets in your area. Which of your three statistics is likely to be

most useful to them?

Let’s Practice:1. Ara Galang keeps a note of the numbers of aces that she

received in successive deals. The numbers are: 0, 2, 3, 0, 0, 2, 1, 1,

0, 2, 3, 0, 1, 1, 2, 1, 0, and 0. Find (a) the mode, and (b) the mean

of the numbers of aces received.

2. The shoe sizes of the members of a football team are: 10, 10, 8,

11, 9, 9, 10, 11, and 10. Find (a) the mean shoe size, (b) the

median shoe size, and (c) the modal shoe size.

3. The marks obtained in a mathematics test marked out of 50

were: 35, 42, 31, 27, 48, 50, 24, 27, 21, 37, 41, 34, 12, 18 and 27.

Find (a) the mean mark and (b) the median mark.

Assignment:

1.What are the levels of

measurement? How do

they differ from each

other?

Levels of Measurement

It is a classification of purported

forms of measurement that

describes the nature of

information within the numbers

assigned to variables.

Reflect:Exactly how the measurement is

carried out depends on the type

of variable involved in the

analysis. Different types are

measured differently.

Levels of Measurement

(1)Nominal Level of Measurement

(2)Ordinal Level of Measurement

(3)Interval Level of Measurement

(4)Ratio Level of Measurement

Nominal Level of

MeasurementIt is characterized by data that

consist of names, labels, or

categories only. The data cannot be

arranged in an ordering scheme

(such as low to high).

Note:

In nominal level of

measurement, we just classify

or categorize the response of

each respondent.

Note:

Nominal scales embody

the lowest level of

measurement.

Ordinal Level of

MeasurementIt involves data that may be

arranged in some order, but

differences between data values

either cannot be determined or

are meaningless.

Ordinal Level of

MeasurementIt involves dichotomous data with

dichotomous values, as well as,

non-dichotomous data which

consists of spectrum of values.

Example 3.7:

A researcher wishing to measure

consumers’ satisfaction with their

microwave ovens might ask them to

specify their feelings as either “very

dissatisfied”, “somewhat dissatisfied”,

somewhat satisfied,” or “very satisfied”.

Interval Level of

MeasurementIt is like the ordinal level, with the

additional property that the difference

between any two data values is

meaningful. However, there is no

natural zero starting point (where none

of the quantity is present).

Example 3.8:

Consider the Fahrenheit scale of

temperature. The difference between

30 degrees and 40 degrees represents

the same temperature difference as

the difference between 80 degrees

and 90 degrees.

Ratio Level of

MeasurementIt is the interval level modified to

include the natural zero starting point

(where zero indicates that none of the

quantity is present). For values at this

level, differences and ratios are

meaningful.

Do you know?

The ratio scale of

measurement is the

most informative scale.

Example 3.9:

The amount of

money you have in

your pocket.

Considerations for Choosing a

Measure of Central Tendency

For a nominal variable,

the mode is the only

measure that can be

used.



For ordinal variables, the mode

and the median. The median

provides more information (taking

into account the ranking of

categories).



For interval-ratio variables, the mode,

median, and mean may all be

calculated. The mean the most

information about the distribution, but

the median is preferred if the

distribution is skewed.

Let’s Practice: Identify the level of

measurement for the following:

1. True or False Test

2. Respondent’s Ethnicity

3. Time of the day ( 7AM to

8AM)

4. Time spend in reviewing

5. Meal Preference (Breakfast,

Lunch, Dinner)

6. Tape Measurement

(Centimeters)

7. Political Orientation

(Republican, Domincan,

Democratic, etc.)

8. Weight

9. Military Rank (Lieutenant,

Captain, Major)

10. Dates (January 14, 2015 to

March 28, 2015)

11. Skin Complexion (Brown,

White, etc.)

12. Likert Scale

13. Years ( 2001, 2002, 2003,

2004)

14. Ruler (Inches)

15. Income (Money Earned Last

Year)

16. Parts of Speech (Noun,

Pronoun, Adjective, etc.)

17. Grade Point Average (GPA)

18. Gender (Female and Male)

19. IQ

20. Hometown (Antipolo City,

Cardona, Rizal, etc.)

Answer:

1. Ordinal

2. Nominal

3. Interval

4. Ratio

5. Nominal

6. Ratio

7. Nominal

8. Ratio

9. Ordinal

10.Interval

11.Nominal

12.Ordinal

13.Interval

14.Ratio

15.Ratio

16.Nominal

17.Ratio

18.Nominal

19.Ordinal

20.Nominal

Assignment:

1.Research about Sigma

Notation.

2.Rules in Sigma Notation.

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Is the problemmigomendoza.weebly.com/uploads/5/4/7/4/54745209/... · Honeygirl Pulot-pukyutan feel hungry and have soup for lunch. She choose one of the 400 grams cans bought in the