ETC1000 Topic 2b

Summary

This instructional video script delves into the nuances of data analysis, emphasizing the importance of understanding and visualizing data effectively. It covers the generation of descriptive statistics in Excel, the creation of frequency distributions and histograms, and the use of box and whisker plots to illustrate data spread and central tendency. The presenter also discusses the significance of probability distributions, particularly the normal distribution, and how to calculate probabilities and identify corresponding data points using Excel functions. The script underscores the necessity of clear and accurate data visualization for effective communication.

Takeaways

• 📚 Watch the first video for Topic Two before this one for better understanding of the material.
• 📈 Excel can generate various descriptive statistics like mean, median, mode, standard deviation, and range with a single step.
• 📊 Visualizations like histograms and box plots are crucial for effectively communicating data distribution and should be clear and well-presented.
• 📉 Numbers alone may not be enough; visualizations help provide a clearer picture of data distribution and are more memorable.
• 🗂 Creating a frequency distribution table helps in understanding the categorization and range of data, such as income levels.
• 📝 Customizing histogram appearance, including labels and bin ranges, is essential for accurate and effective data representation.
• 📉 Box and whisker plots are useful for visualizing quartiles and understanding the spread and central tendency of data.
• 📊 The choice of bin ranges in histograms can significantly affect the interpretation of data, as demonstrated with grade distributions.
• 🤔 Probability distributions, including the normal distribution, are foundational concepts for understanding how random data is distributed.
• 📝 The normal distribution is common in various fields because it represents data that is symmetrically distributed around the mean with probabilities decreasing as values move away from the mean.
• 🔢 Excel's norm.dist function can calculate probabilities for a normal distribution or find the x value for a given probability.

Q & A

What is the main focus of the second video in the series?

The main focus of the second video is to complete the discussion on the second topic, which involves finishing off the rest of the material after introducing the concept of standardizing data and summarizing its characteristics using measures with quantitative data.

Why is it important to watch the first video before the second one?

It is important to watch the first video before the second one because the second video continues from where the first left off, and concepts introduced in the first video are built upon in the second, ensuring that the content makes sense and is understood in the correct sequence.

What are some of the descriptive statistics measures discussed in the video?

The descriptive statistics measures discussed in the video include mean, median, mode, standard deviation, variance, range, minimum, and maximum.

How can one obtain all the mentioned descriptive statistics in Excel with a single step?

One can obtain all the mentioned descriptive statistics in Excel with a single step by using the 'Descriptive Statistics' tool, which provides all these measures at once, although it may not include all measures like quartiles and interquartile range.

What is a frequency distribution and why is it useful?

A frequency distribution is a table that shows the number of data points that fall within certain ranges or categories. It is useful because it provides a detailed view of the data, showing how many observations fall into each defined range, which helps in understanding the distribution of the data.

How does the video script emphasize the importance of visualization in data analysis?

The script emphasizes the importance of visualization by stating that visualizations can be more memorable and clearer than numbers alone. It also mentions that the presentation of data through graphs can be misleading if not done properly, highlighting the need for careful and accurate visualization to communicate the data effectively.

What is a histogram and how does it help in understanding data distribution?

A histogram is a graphical representation of the distribution of data. It groups data into intervals, or 'bins', and shows the frequency of observations within each bin. It helps in understanding the data distribution by visually showing the concentration of data points, the spread, and any patterns or outliers.

What is a box and whisker plot and how does it represent data?

A box and whisker plot is a standardized way of displaying the distribution of data based on a five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. It provides a clear visual representation of the spread and skewness of the data, including potential outliers.

Why is the normal distribution important in statistics and what does it represent?

The normal distribution is important in statistics because it is a symmetrical bell-shaped curve that represents the distribution of many real-world phenomena. It is widely used in statistical inference and hypothesis testing due to its properties and the central limit theorem, which states that the sum of independent and identically distributed variables will be approximately normally distributed regardless of the original distribution.

How can Excel be used to calculate probabilities associated with a normal distribution?

Excel can be used to calculate probabilities associated with a normal distribution using the NORM.DIST function, which calculates the cumulative probability for a specified value in a normal distribution with a given mean and standard deviation. The function can also be used in reverse to find the value (x) that corresponds to a given probability.

What is the significance of standard deviation in the context of the normal distribution?

In the context of the normal distribution, the standard deviation indicates the spread of the data around the mean. A smaller standard deviation means the data points are closer to the mean, while a larger standard deviation indicates a greater spread. It is also used to define the ranges within which a certain percentage of data falls, such as within one or two standard deviations from the mean.

Outlines

00:00

📈 Descriptive Statistics and Data Visualization

This paragraph introduces the importance of watching the first video on topic two before proceeding and promises a faster pace for the current video. It discusses the process of standardizing data and using Excel for descriptive statistics such as mean, median, mode, standard deviation, variance, range, minimum, and maximum. The speaker also touches on the limitations of Excel for certain measures like quartiles and interquartile range, suggesting that these will be covered in future lessons. The focus then shifts to the value of visualizations, such as histograms, for better communication and understanding of data distribution, emphasizing the need for clear and accurate presentation to avoid misleading interpretations.

05:02

📊 Histograms and Box and Whisker Plots for Data Distribution

The speaker explains how to create a frequency distribution table and a histogram using Excel to visualize income data. They highlight the need for adjusting the histogram's appearance for better clarity and communication, including closing gaps between bars and adding labels. The paragraph also introduces the concept of cumulative percentages and percentages to provide a clearer picture of income distribution. Additionally, the speaker mentions the box and whisker plot, which uses quartiles to summarize data distribution, and emphasizes the importance of visual presentation in data analysis.

10:03

📉 Box and Whisker Plots and Histogram Improvements

This paragraph delves deeper into the box and whisker plot, discussing its components like the median, quartiles, and outliers. The speaker uses an example of student grades to illustrate how the plot can convey a lot of information about data distribution in a single graph. They also address the limitations of Excel's automatic histogram generation, demonstrating how to improve the histogram by specifying bin ranges and adjusting the presentation to better communicate the distribution of grades, such as aligning bins with grade boundaries.

15:04

📚 Probability Distributions and the Normal Distribution

The speaker transitions to theoretical concepts, starting with the idea of random data and probability distributions. They explain how to represent data as probabilities and the characteristics of a probability distribution, such as being mutually exclusive and exhaustive. The paragraph introduces the normal distribution, a bell-shaped curve that is common in various fields due to its symmetrical nature and the likelihood of values close to the mean. The speaker also discusses the standard normal distribution and how to calculate probabilities associated with it using Excel functions.

20:04

🔢 Understanding the Normal Distribution in Excel

The paragraph focuses on how to work with the normal distribution in Excel, using the NORM.DIST function to calculate probabilities for given x values and the NORM.INV function to find x values for given probabilities. The speaker illustrates this with examples, such as finding the probability of a value being less than zero in a standard normal distribution and determining the x value that corresponds to a 97.5% probability in a distribution with a mean of 0 and a standard deviation of 1. The importance of understanding these calculations for future topics is emphasized.

25:06

📘 Conclusion and Encouragement for Tutorial Practice

In the final paragraph, the speaker thanks the audience for their patience and perseverance, expressing hope that they enjoyed the topic and will learn a lot from the tutorial and other exercises. The speaker encourages continued practice with Excel and the normal distribution, setting the stage for future topics where these concepts will be applied.

Keywords

💡Standardizing

Standardizing refers to the process of transforming data into a common format to make it comparable. In the context of the video, it is the initial step taken to prepare data for analysis. The script mentions that before any analysis is done, data should be standardized to ensure it is in a comparable form.

💡Descriptive Statistics

Descriptive statistics are numerical measures that summarize and describe the characteristics of a data set, such as the mean, median, mode, standard deviation, and range. The video script explains how Excel can be used to calculate these measures in one step, providing a quick overview of the data's central tendency and dispersion.

💡Quantitative Data

Quantitative data represents values that can be measured numerically, such as income or age. The video script discusses measures used with quantitative data to summarize its characteristics, emphasizing the importance of understanding where the data is centered and how spread out it is.

💡Frequency Distribution

A frequency distribution is a table that shows the number of data points that fall within specified ranges or categories. The script describes how to set up income ranges and use Excel to determine how many people fall into each range, providing a detailed breakdown of the data.

💡Histogram

A histogram is a graphical representation of the distribution of data, with bars representing the frequency of data points within certain ranges. The video script explains how to create a histogram in Excel, emphasizing the importance of presentation to make the data clear and visually appealing.

💡Visualization

Visualization refers to the use of graphical representations to communicate information effectively. The video script stresses the importance of visualization in data analysis, noting that a well-presented graph can be more memorable and informative than raw numbers.

💡Box and Whisker Plot

A box and whisker plot is a standardized way of displaying the distribution of data based on the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The script introduces this plot as a useful visualization tool that can effectively communicate the spread and skewness of a data set.

💡Quartiles

Quartiles divide a data set into four equal parts, with the first quartile (Q1) representing the 25th percentile, and the third quartile (Q3) representing the 75th percentile. The video script discusses how quartiles are used in box and whisker plots to summarize the distribution of data.

💡Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is characterized by its bell-shaped curve. The script explains the characteristics of the normal distribution, such as its symmetry and the likelihood of values occurring close to the mean, and how it is commonly found in natural and financial phenomena.

💡Probability Distribution

A probability distribution is a statistical description that shows the likelihood of different possible outcomes within a data set. The script describes how the table of percentages for income ranges can be considered a probability distribution, where each category represents the likelihood of a randomly chosen individual falling into that income bracket.

Highlights

The importance of watching the first video for topic two before the second for better understanding.

Excel can generate descriptive statistics like mean, median, mode, standard deviation, variance, range, minimum, and maximum in one step.

Some statistical measures like quartiles and interquartile range are missing in Excel's one-step descriptive statistics.

Visualizations are as important as numbers for summarizing and communicating data effectively.

Creating a frequency distribution table to understand income ranges and the number of people in each range.

Transforming raw numbers into a histogram for better visualization of data distribution.

The necessity of refining initial visualizations for clarity and accuracy in communication.

Using percentages to provide a clearer picture of income distribution among people.

The introduction of box and whisker plots for visual representation of quartiles and median.

Interpreting box and whisker plots to understand the distribution of student scores in a class.

The impact of choosing the right bin ranges in histograms to effectively communicate information.

Calculating probabilities using the normal distribution and Excel's NORM.DIST function.

Understanding the concept of standard normal distribution with a mean of zero and a standard deviation of one.

Using Excel to find the x value that corresponds to a given probability in a normal distribution.

The significance of probability distributions in representing the likelihood of different outcomes.

The characteristics of a probability distribution: mutually exclusive and exhaustive.

The prevalence of the bell-shaped normal distribution in various aspects of life and its importance in statistics.

Transcripts

hello everybody

topic two second video

make sure you make sure you watch the

first video for topic two

uh before you

watch this one otherwise things won't

make quite so much sense to you

we're going to finish off the rest of

the second topic

now uh hopefully i will go a little bit

quicker apologies for the last video

being a bit too long for you but

hopefully this one will be zooming along

a bit faster

we spent most of the last video after we

talked about standardizing getting the

data in a comparable form before you do

anything we talk most of the time about

measures which are used with

quantitative data to sort of summarize

the characteristics of the data where is

it centered how spread out it is and

what kind of shape it takes

as it turns out we can get uh

almost all of the measures we're

interested in with one step in excel and

i'll show you the steps for how to

produce this descriptive statistics

thing uh later but essentially you can

get the mean and the median and the mode

and the standard deviation the variance

all those things that we talked about

the range the minimum max all those

things we talked about

uh in one go and that's kind of

convenient and i'll show you in a little

while how to do that there's a few that

are missing like the quartiles in the

interquartile range and there's a few

that we haven't taught you which you can

worry don't worry about until future

subjects if you learn some more

so that's just a little reminder or a

footnote for you

all of these things though as the others

are that i introduced in the previous

video are

numbers that summarize things and

numbers are pretty useful way of

summarizing things if you want to say

what's the average income of people in

this area thirty thousand dollars per

year is a pretty useful number to give

people uh it's for example what are the

range of incomes

dollars zero dollars up to six hundred

one thousand so numbers are valuable but

often visualizations are also good

so for example we might want to produce

some kind of uh

distribution some picture that gives us

an idea about the distribution of the

data so to do that

first of all we might start with that

with a table of numbers so instead of

just giving us for example in this case

here we've got uh information about

incomes and we've got it we know where

the data is centered we know how spread

out it is according to the standard

deviation of 35 000 per year etc but

what we'd like perhaps is is sort of a a

bit more detail about the sort of

categories and the ranges so we set it

up in what's called a frequency

distribution so we define some income

ranges

naught to ten thousand ten to twenty

thousand and so on and then we ask excel

to tell us how many people earn

an income in this particular range so

1509 people earned less than ten

thousand dollars 939

people earned between 10 and 20 000 and

so on okay

and 162 people earned more than a

hundred thousand dollars that's

basically what this table gives us

of course this is a pretty ugly

distribution if you wanted to put this

in a nice report you'd give it some

better labels than i've given there and

put dollar signs on the

bins and so on but

this is how it's dumped to us when excel

gives it to us it's a starting point of

the raw numbers

but better still let's show those

numbers in some kind of graph for

histogram

and so i'll show you in a couple of

minutes how i produced this table in

this graph but for now

i'll just show you the output of it i've

created a histogram and i've played

around with the presentation of the

histogram to make it look nice i've

given it nice headings and labels and

i've also closed up the gaps between the

bars for reasons

but i'll explain in a moment

important stuff that you have to do in

order for this final product here to

make sense visualization

is about communication

you might think as a data analytics nerd

that

how it looks is not that important it's

the substance well it couldn't be

further from the truth

how things look

is extremely important because firstly

people will remember it if it's

if it's clear secondly you can make it

look

in such a way that it's misleading to

people you can present things in a

certain way that's not giving a true

picture

or you can do the opposite you can

present it in a way that gives a clear

picture about what's going on people

will remember the message of a picture

far better than they'll remember a whole

bunch of numbers

so we spend quite a lot of time getting

this histogram right not just dumping

the first histogram that comes out of

exception so i really want to emphasize

that

that if you're ever asked to produce any

kind of visualizations

the first visualization you get out of

your software is going to be pretty

might have the right numbers in it but

it's going to be pretty ugly and you're

going to need to spend a good amount of

time

going through all the other steps of

making it look right before you finish

the task

and that might be 80 of the time making

it look right 20 of the time getting the

first version so i can't underestimate

that

so you see here that we've got most of

the people earning less than ten

thousand dollars not actually most of

the people but the biggest most common

category here is that and then as you

get further and further up in the income

range less and less people are in each

of those ranges and then there's a bit

of a blip at the end of people who are

into more than a hundred thousand

dollars okay so now i've got a bit more

of a picture about the sort of

distribution of incomes it's quite a

different picture to what i learned from

just looking at the average the mean i

think was thirty seven thousand dollars

somewhere in around about here so sure

the average income is thirty seven

thousand but a whole bunch of people are

earning less than that including some

people that are earning less than ten

thousand so not very much at all this is

not uh australian data by the way this

is from from the us

unfortunately in certain parts of the us

there's a large number of very poor

people who are in very little

one of the

richest developing countries in the

world

strange thing to say but think of that

now sometimes it's good for us to not

just give the numbers but actually to

give them percentage terms so i've asked

excel to produce the

cumulative percentages and then i've

turned those into actual percentages by

subtracting the difference you'll get a

chance to play with this sort of thing

in your tutorial work and so now instead

of saying there are 1509 people who are

in this range i can say

nearly percent of people earn less than

ten thousand dollars you see that's

pretty neat or

just over three percent of people earned

more than a hundred thousand dollars

or five point six six percent of people

earned between fifty and sixty thousand

dollars that's perhaps not as

interesting or i can look at the

cumulative numbers and i can

for example say

okay so let me just highlight what i'm

looking at here

uh

so i've i've talked about this group

here 29 and less than 10 000 3

earning more than 100 000 or i might go

over here and say 91 percent of people

earn less than 70 000

okay so all of these numbers are quite

useful they're giving us a bit of a

picture about how the distribution

actually looks and then i might want to

also perhaps

change my histogram that i had on the

previous thing and instead of the

bars up here being number of people they

might be percentages based on that

column there okay so there's a few ways

in which i'm buried to make it better

and some other suggestions there making

sure your boundaries are right and so on

so i won't go through that in detail

i'll let you read that and absorb it but

i think i've got the message across it

really has to look good if it doesn't

look good then it's bad okay and it's

not acceptable

um

one other graph of visualization i'm

going to

introduce you to is the box and whisker

plot and we'll have a look at it and how

we present that in a moment and the

reason i like the box and whisker plot

is because it works off those quartiles

the median the mean the q1 and the q3

that we talked about in the previous

class

remember in the previous

video i introduced you to some results

from a first-year stats

unit

one of

your units that you're studying from a

previous year and here we had the

summary data for that particular unit so

63.8 was the mean the median was 65 so

half the students got less than 65 half

the students got more

the standard deviation was 15.3 so that

was so roughly speaking the average

variation from the mean

the first quartile was 58 core of the

students got less than 58.

a quarter of the students got more than

74

so the middle 50 percent got somewhere

between 58 and 74 a interquartile range

of 16

that's pretty neat way of describing it

but even better if i could describe that

with something visual so let me show you

how i produce

okay so there's the data that i've just

been showing you there the ranges i'm

going to produce something called a box

and whisker plot which is what this

thing here looks like excel gives you

one of these and i and uh

i explained briefly in the notes there

how you

find it under the menu um

insert the excel

charts and uh look under the histogram

button you'll see box and whisker plots

okay

now the box and whisker basically this

solid blue bit is the the middle 50 so

q1 the first quartile

is

58

up to 74 that's q1 to q3 and 65 is your

median so those three numbers there are

the bottom 25 percent cut off the 50 cut

off in the 75 cup so that's pretty

useful that's where

if i look at my class half my students

get a score in that range there okay

i've got a few really top students who

score a better than that and my

top number is 91. now that's not the

very top mark

uh

in every case in this case it is

actually the very top mark

the box and whisker plot produced by

excel

make some decisions using some

algorithms which we won't go to the

details of as to whether it's shooting

that the the

range from 91 to the top sort of

line to the bottom line should be from

their maximum to the minimum or whether

it should treat some of the numbers as

what's referred to as outliers now this

is an interesting example

it didn't find any outliers at the top

but there's no totally weird student who

got 98 percent here the top mark was 91.

at the other end according to excel's

algorithm it said look

a reasonable range of the data is 34 but

there's actually sort of about eight

students who got really low marks who

sort of don't count shouldn't be treated

in the data they're outlines

now that's excel's judgment and we don't

get any control over that because this

is a fairly simple piece of statistical

software but if when you learn

programming uh in r and future units

you'll be able to produce much better

box and whisker plots than this but

basically this is telling us

half the data got half the students got

between 58 and 74. the top mark was 91.

so

the range of almost all the students was

between 91 and 34

but there was a very small number

eight or so students who got actually

even further below that so but out of

the 600 or so students in this class

that's a very small sample so it gets

wrapped down the bottom

and so that's how i interpret that box

and whisper plot and i think that in one

graph there i've got quite a lot of

information if i had my choice i

probably wouldn't do those little little

dots down the bottom uh i'd make my plot

look a bit neater but it's good enough

why is it called a box and whisker well

that's the box and these little lines

the 91 and the 34 they're the whiskers

you know if you're someone with a

moustache and little whiskers on the end

that's what that is referred to if you

try to figure out the jargon

all right so

uh

there's one okay i might want to produce

a histogram of this data but actually if

i do a histogram

it's actually not a very good one let me

let me do it quickly now to give you an

idea about what i mean so i'm going to

go to the data menu

and i go to data analysis so in the data

menu up here i go to something called

data analysis and hopefully you've got

that and i ask for it if you don't then

you need to do some add-ins

for analysis full pack which you can

learn about in your tutorial

uh then i go to histogram and i say okay

and i'm going to send the output to some

new worksheet just so that i get mess

and i'm going to make sure i ask it for

a chart

let's see what comes about what's i

better tell it where my data is there it

is there so it's uh row

eight two

a five eight six i just happen to know

that from memory

okay it's calculating the histogram

oops

here it is okay

what an absolutely ugly histogram

okay

it's terrible

that's exactly what you'll get if you

just ask excel to give you a histogram

what a load of rubbish there's so many

things wrong with that histogram

so i really don't like it at all okay

and you can see why it's pretty obvious

okay the data ranges are all stupid um

yeah we don't even need to waste our

time talking about why that's bad we

need to make a better job of it than

that and so read the notes there and

figure out how to do it and in

particular in this particular case i

actually

decided that well first of all the first

thing i need to do is actually

i need to specify the ranges of the bins

so when i did my histogram i was

actually i had a choice then i chose not

to take it if i go to data analysis

histogram again instead of just

leaving that bin range if i leave the

bin range blank

it will just choose ranges for me and it

chose stupid ones so instead of that

maybe a simple improvement i can make is

let's let's use these numbers here

and use those ranges and see what

it looks like well now it's looking a

lot better isn't it that's much nicer

looking histogram i've sort of got and

in fact uh

for reasons that i'll get to in a moment

i've chosen these ranges very carefully

okay so that's an immediate improvement

i can make and then there's still more

that i need to be able to do to that

but actually the choice of this range

i'll go back to the other

sheet here this is

the final histogram i produced after

mucking around a bit further

and i've done a few things there

first of all i chose this particular

range of bins for a very good reason

namely this is the grades that you

achieve in this unit if you get 49 or

less you get an n if you get 59 or less

you get a p

and 69 is a credit 79 or less is a

distinction and above 79 is a higher

stitch okay

so i chose to do my histogram not with

equal space bars or anything but

actually to communicate what is most

interesting and relevant to students

namely what grade are you going to get

here or what's the distribution of the

grades

okay so that's what i ended up doing

there

in order to produce histogram this

histogram i changed the the default is

the bin ranges would have come out as

49.59 but if i just change the type

over those with np etc then i can end up

with much nicer labels in my histogram

likewise the frequencies that were here

were

the original data

the account of how many people are in

each case each category like over here

you'll see here here we are you see the

bin the values are 89

but i don't want that i want the

percentage of people who got or the

proportion of people who got each

value well that's fine i just took the

89 and i divided by however many data

points i had 583 and the next one 72 and

i divided by 583 so i can change these

numbers once excel's produced them for

me and get a nicer looking histogram so

on this axis i've now got the proportion

of students who got that score so you

see here 34 35 34 of students got a

credit

etc uh 13 because 12 of students got a

high distinction 15 of students got to

fail etc okay so i worked quite hard 80

of my time was spent making

that ugly looking histogram there into

that nice looking histogram there okay

and importantly i chose to do it in a

for the purpose okay

what's the question i'm trying to answer

and how does my

visualization best answer that question

and what communicate that information

and in this example that specifically

had to do with choosing the categories

so that they corresponded to the grades

that students achieve in the unit but

obviously in other contexts it's going

to be a different set of issues you have

to address to get the best possible

visualization

okay

now

switch gears that's all been practical

how to show stuff in data and

particularly today how to visualize i'm

going to take us now to a little bit of

what you might describe as theory and

the reason i'm going to do that is

because we want to have a deeper

understanding of the underlying

principles behind all of these methods

that we use and the deeper understanding

is with the idea that we have random

data and the random data is distributed

according to probability distributions

so you did a little taste of probability

in topic one we're going to do a little

bit more on probability here in topic

two and more in later weeks just so that

you get more comfortable with thinking

about random data and about probability

associated with outcomes

we can think about the the table of

numbers that we've produced the

percentages of people in each

income range for example as a

probability the probability if i chose

someone randomly this is the same table

that i had earlier in the present in the

in the topic but now i've just turned

these numbers here instead of how many

people earn between north and ten

thousand dollars i've turned it into a

probability a proportion

now i can say if i chose someone

randomly

from this suburb what's the chances that

they'll earn between north and ten

thousand dollars answer 0.29 600. that's

a probability that's what we mean by

probability the probability of something

happening

and you'll see the probability that

they'll learn more than a hundred

thousand is far less okay so think about

these numbers not just as however many

people are in that code agreement think

about them as probabilities

notice there's a couple of

characteristics of this thing here that

make it a probability

distribution they're mutually exclusive

you can't earn between 30 and 40 and

between 40 and 50. everybody is there

only once

so you can't be in more than one cabin

and secondly

it's exhaustive everybody's there if i

add these probabilities up i get one

somebody has to earn

something between naught and a million

dollars or an infinite bolts okay

so it's mutually exclusive and it's

exhaustive and that's essential for this

to be classified as a probability

distribution

with some data

you can present the probability

distribution in a table because there's

only a limited number of values it can

take in this case we've categorized it

so there's only like 10 or 12 categories

here

10 or 11 categories other data you might

have it in the form of that of a

probability density function which is

like a smooth curve and we're going to

see an example of that now to finish off

this topic and that's called the normal

distribution

what's the normal distribution about

well you've probably come across this

before in high school

and it's a sort of a bell-shaped

distribution that looks like this and

the reason it's so common and so popular

is because think about this as a range

of possible values of some variable x

could be how much income people earn or

price of houses or

you know how many

children a person has

that's probably not a good example

because the range of that's pretty small

but you know

things that can vary a lot it's a whole

possible set of values that that

variable can take

it's very common for

the probabilities associated with

different possible values of that

variable to follow this bell shape the

reason it's common is that the mean is

in the middle

and if

values that are close to the mean

are much more likely to occur you know

the probability is given by the height

of the bar just like a histogram

the high histogram points are the ones

that are more common so these are the

range of values of x which are much more

likely to occur and they're close to the

mean

the further you get from the mean the

muscle much less likely it is that

you'll get those kinds of values

in either the negative direction or the

positive direction and that's

essentially what describes a bell-shaped

curve

in addition to that it's symmetric

whether you're going out in the positive

direction or the negative direction you

get about the same sort of decrease in

probabilities of it occurring

and that's not true of every set of data

in the world i've already given examples

of skewed data like house prices and so

on but actually a lot of data is pretty

close to normally distributed

it's got those characteristics have been

symmetric most values close to the

values close to the mean are much more

likely to occur and as you get further

and further from the mean

so that's why we study it because in

natural world and in financial markets

and in

business

any economic development and economies

data often ends up looking quite

normally distributed

okay uh this is a particular type of

normal distribution it's it's it's

what's called the standard normal

because it has a mean of zero

so the average is zero and it has a

standard deviation of one

which means that if as i go out a couple

of standard deviations it becomes very

unlikely you'll get values

out here and when i get to three

standard deviations above the mean i've

got virtually no chance of that

occurring and by four standard

deviations it's infinitesimally small

and likewise so these can be sort of

thought about as like numbers of

standard deviations above or below the

mean

let's take the heights of adults okay

you know in a country okay there'll be

an average height here for males of 1.8

meters or something like that and then

as you go

one standard deviation the standard

deviation of heights might be about five

centimeters so as you go up one one

standard deviation to 1.85 meters

it's less likely you'll meet someone

that tall it's even less likely you meet

someone 1.9 meters tall 1.95 meters 2

meters etc

and likewise 1.75 meters is less likely

than 1.8 1.7 meters is even less likely

1.65 is even less likely 1.6 metres even

less since we're talking particularly

about male adults here okay so

that's exactly the sort of a very

commonplace distribution that's occurred

now

i can work out probabilities associated

with normal distributions using excel

and

in normal distribution it's it's applied

in cases where data can take all sorts

of values so we don't normally talk

about what's the chances of someone

being exactly 1.83 centimeters tall we

actually prefer to talk in ranges what's

the chances of somewhere between being

between 1.8 and 1.85

or above 1.83 or something so those are

the kind of probabilities we can work

out and we do that by calculating the

area under the curve of this nice

bell-shaped thing so the probability of

of someone uh

of a height more than two standard

deviations above the mean would be this

shaded area here

and so we can calculate that probability

or the probability of for a distribution

which has got a mean of 10 and a

standard deviation of 3 the probability

of someone

having a value of less than 5 is this

shaded area here ok

how do i calculate those probabilities

well if i want

to work out the probabilities i just

have to use a function in excel called

norm.dist so if i've got a normal and

the idea is this is let me just show you

what these different components of it is

that you'll get to do this in your

tutorial work so i won't go into the

detail here that's the mean so i've got

a normal distribution with a mean of

zero

that's the standard deviation it's got

us

ah sorry no i'm wrong that's the normal

that's the probability for which i want

to calculate the x value i want to

calculate for so i want to calculate the

chances of x being less than zero

okay

and i've got

a normal distribution with a mean of

zero so that's the mean

and that's the standard deviation

and true just means i want the

probability of being less than it

so that's actually precisely

excuse my terrible normal curve that's

the standard normal and i want to work

out the probability being less than zero

so i'm actually working out that

probably there

surprise surprise it's a half

remember it's symmetric the probability

of being less than naught

is exactly equal to the probability of

being greater than not 50 or 0.5

or i could take my normal distribution

with a mean of 0 and a standard

deviation of 1 and i could look at say

what's the chances of it being less than

some number up here like 1.96

excuse my i can't write very well with

my opinion but you know that's 1.96 then

that's why you're going to see

well that's a

quite a large probability because it's

all of this area here from way so where

there by now it works out to be about

97.5

chance

okay

that's what we can do in excel or we can

do the reverse we can say if i give you

the probability can you tell me what the

x value is

so i say

i've got a standard normal a normal with

a mean of 0 and a standard deviation of

1. and i want to know what's the x value

that gives me a probability of 0.5 of

being below it answer well that's the

reverse of the question we just asked

before zero half the values

are less than zero or if i've got a

normal distribution

uh with a mean of zero and a standard

deviation of one and i want the

probability to be 0.975 what's the x

value answer 1.96 okay so i'm just doing

the same thing as before but in reverse

okay so that's how we do those things in

excel for now park that practice it a

little bit and we'll get to make use of

it in future

topics when we actually start applying

the non-distribution

okay

thank you for your patience and

persevering with that topic i hope you

enjoyed it and i hope you can learn lots

as you work through the tutorial and

other questions