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Mean Median Mode

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This content introduces fundamental statistical concepts for describing data, focusing on measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation), along with frequency distributions and histograms.

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we are going to discuss this describing data

data

with your numerical measures

so our goal is to calculate the

arithmetic mean

weighted mean median mode and geometric mean

mean

and then we can explain the

characteristic uses the advantages

and disadvantages of each measure of locations

locations

and we can identify the position of the

mean median

and mode for both symmetric and good distributions

distributions

and we can compute and interpret the

range mean division variance and standard

standard

deviation okay first

we're going to discuss the frequency

distribution this is a table that

organizes the data values in

into classes on intervals along with number

number

of of values that fall in each class

so it this will be your frequency summation

sun group frequency distribution

for data sets with few different

values each value is in its own class

for the group of frequency distribution

for data sets with many different values

which are grouped together in

classes so if you adding

30 we have 202 i'm frequenting yeah

so 32 42 we have one 508

43 and 50 54 so we have 620

for 55 to 66 we have 413

67 to 78 is 158

78 to 90 is 32.

difference category we have ungroup

and group and then for example

for the ungroup frequency distribution

one

four and five so we have five

ah so four one

two three

four five six

seven eight and nine adiva

so gonna think it's a five in eight and

so for the frequency histogram a bar

graph that represents the frequency distribution

distribution

so the horizontal scale is the quantitative

quantitative

and measured the data values yes horizontal

horizontal

unlike this vertical iteration measures

that non-frequency of classes

frequent and just a five month frequency

so four

adjectives are nine and so five

okay next so it's a relative

frequency distribution shows shows the

proportion of percent or percentage of

the data

that falls in a particular class for

example we have relative frequency is

equal to

class frequency over the sample size

so relative frequency histogram has the

same shape

and the same horizontal scale as the

corresponding frequency

histogram so the vertical scale

measures the relative frequencies not

frequencies relative frequencies

not the frequencies okay so

has the same shape and horizontal scale

as a and as a histogram but

the the vertical scale is marked with

relative frequencies so get reading

samples so

this will be our relative frequency

30 35 percent three percent and five

percent so it's

a thing a sample started

for the data sets with many different

values for the group frequency distribution

distribution

so group theta into 5 to 20 classes of equal

equal

with you know fdm frequency n

okay for example yeah it's got exam scores

scores

40 to 4959 in london 69 in london

and then 7279 illandin

and so on and so forth for example

so 30-30 is a 40-49

0. there is a 50-59 up but

nine 66-69 seven and thirteen and diamond

so eighty to eighty one nine is we have

ten then nineteen to ninety nine we have three

three diva

for the lower class limits are the

smallest number

that can actually belong to the

different classes

for the upper class limits are the largest

largest

numbers that can actually belong to

different classes

class feed is the difference between two

or later on okay okay

so the class midpoint actually the value

halfway between

lower class limit and then the upper

class boundaries demand class halfway between

between

an upper class limit and the next lower

upper class limit plus the next lower

class limit over to

okay so what are the characteristics of

of the mean so the arithmetic mean is

the most

widely used measure of location it may require

require

requires the interval scale its major

characteristics are all values are used

the sum of deviation from the mean is zero

zero

it is unique and it is

calculated by summing the values

and dividing the number of values actually

so we have two types of means so we have

the population mean and then the sample mean

mean

for the population mean for ungroup data

the population mean

is the sum of all the population values

divided by the total number of population

population

values so we have population mean so

summation of

x over and where x is representing any

particular value

mean nothing will be the number of the

values in the population okay for example

example

so there are 12 automobiles

manufacturing companies

automobile manufacturing companies in

the united states

so listed below is the given the number of

patents granted by the united united states

states

government to each company in a recent year

year

so we have the company number of patented

patented

so in this in this information

a sample or a population

in population so what is the arithmetic mean

mean

of a number of patents

five one one plus three eight five plus

two seven five plus two five seven

plus two three four plus two ten plus

ninety seven plus fifty plus thirty six

plus twenty three

plus thirteen so no homogenous

so our population mean is 195.

okay for the sample mean for group data

the sample mean

is the sum of all the sample values

divided by the number of

sample values sample mean is the sum

x or bar x will be the sample mean

and will be the number values in the

sample for example

globe telecom is studying the number of

minutes used monthly

by clients in a particular cell phone

rate plans or a random sample of

12 clients showed the following number

of minutes used last for

last month okay so we have 1991

77 110 94 92 89 100 119

one one 113 eight three minutes

so what is the arithmetic mean number of minutes

minutes used

group for example means so you have to

get for the midpoint

so get the multiplied by the frequency

and then after i know we have used the

total number of the frequency and we

will get the sample mean

okay check that in okay so we have the

frequency this distribution for the vehicle

vehicle

selling prices the information is

repeated below

so determine the arithmetic mean vehicle

selling price

so in millions of million

0.7 is equal to

two e d divided that is a two so

that is zero point six and

zero point six zero point nine one point

two one point three

one point four two point one and two

point four

so it okay so we have solved for the fx

so in frequency multiplied that in 0.6

it takes 4.8

23 times 0.9 20.7

seventeen times one point two twenty

point four

eighteen times one point five is twenty

seven eight times one point eight is

fourteen point four

four times two point one is eight point four

four

two times twenty two point four is four

is one hundred point five so parasol

mean sample mean is one hundred

point five divided that is 80 so we have 1.26

f x x next

so the properties of arithmetic mean

so every set of interval level or ratio

level data has a mean all the values are included

included

in computing the mean a set of data has

unique mean the mean is affected

by unusually large or small data

values the arithmetic mean

is the only measure of central tendency

where the sum of the deviation

of each value from the mean is zero

so next is [Music]

midpoint we're preparing for the

midpoint of the values after they have

been ordered

from the smallest to largest so there's

there are as many values above the

median as below

it in the the data array

so for an even set of values the

median will be the arithmetic average of

the two middle

numbers for example

in properties not in monopoly so there

is a unique

median for each data set it's not

affected by extremely large or

small values and it's therefore a valuable

valuable

measure of central tendency when such

values occur

so it can be computed for ratio level

inter interval level or ordinal level data

data

so it can be computed for an open

ended frequency distribution if the

median does not lie

okay so the age for the sample of five colleges

colleges

so the height of four basketball player

is in each s

are 76 737 80 and

75 so 73 75 76 and 80.

observation that appears most frequently okay

okay

okay next sample

so the data entry that occurs with the

greatest frequency

if no entry is repeated the data set has

no mode

if two entered entries occur with the

same greatest

frequency each entry is a mode

oh by mode you know by modal

okay zero pi point four one point one

zero point four to zero point four

seventy three zero point four eight and

1.1 so

the one base has been getting one point

one so it doesn't move nothing

yeah i'll determine 27 27 27 25 55 55 88 89

so one two three one two three so you know

so 27 and 55 sodito

one two three six seven eight nine ten

okay so the annual salaries of an

architect in selected companies

in the philippines are shown below so

what is the model

okay so the relative position of the mean

mean

so panini my young dream so the range is

the difference between the lowest value

and the highest value so the number of

cappuccinos sold at the starbucks

location in the sm caballon city between

4 pm to 7 pm

for example five days last were 20 40 50

starbucks so we have to determine the

mean division

for the number of cappuccinos sold okay so

range largest value minus the smallest

value so the largest value is 80 the

smallest is 20.

negative 30 40 minus negative 10

50 minus 50 0 60 minus 50 is

okay so next so here

uh the barrier samantha sommer

determined the number of tropics

citation issued during the last five

so 38 26 13 21 and 22.

divide five is 28 so 38 minus 28 is

positive 10

negative 2 negative 15 positive 13

negative 6.

so the hourly wages for the example of

part-time employees at

home depot coming to one are 200 to 110

to 250 and 300

236 is negative 36 negative 26 negative 16

six seven six two five six one nine six and

and

20 over five minus one so we have

one six thirty so you know ating sagorilang

rain x for the standard deviation of

group theta

so this is the formula standard

deviation this is one square

of summation of frequency multiplied

quantity of x

minus the mean raised to the squared all

over n minus one okay

so these are the ibis opinion midpoint

0.6 minus the midpoint negative 0.66

0.9 minus 1.26

is negative 0.36

1.2 minus 1.26 0.06

1.5 minus one point eight is positive

zero point twenty four

it's a positive zero point five f five

four but i mean positive zero point

eighty four

positive one point fourteen so i think meaning

meaning

zero point one two nine six zero point

zero thirty six

zero five seven six zero point two nine

one six zero point seven zero five six

and one point two nine nine six

three and now in baltis multiply nothing

and frequency so eight times zero point

four three five six

three point four four eight it's 23

multiplied zero point one two nine sixty

two point

nine eight zero eight seventeen multiply

zero point zero zero three six zero

point zero

six one two eighteen times zero point

zero five seven six one point zero

three six eight eight times zero point

two nine one six is two point three

three two

eight and four times zero point seven

zero five six two point eight two two

four and then two times one point two

nine nine six

root of frequency frequency multiplied

by a quantity of x minus

mean raised to square root all over n

minus 1.

so we have 15.318 over 80 minus 1

so we have 0.44 here so

okay next so that will be our

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