IB · Mathematics Applications & Interpretation (AI)
AISL 4.3.1 Averages and spread
Learn AISL 4.3.1 Averages and spread in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.3—Mean, median, and mode.
Learn AISL 4.3.1 Averages and spread in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.3—Mean, median, and mode.
Learn AISL 4.3.1 Averages and spread in this free IB Mathematics Applications & Interpretation (AI) video lesson for SL 4.3—Mean, median, and mode.
The video provides a comprehensive overview of basic statistical measures using the example of economics grades from 10 students. Key concepts covered include:
Additional statistics discussed include the range (), lower quartile (), upper quartile (), and the (). The video also touches on and , explaining that variance is the square of the standard deviation, calculated as .
00:00Hi everybody. So in this
00:02lesson I'm going to do
00:03all these things together. The
00:04mean, the more the median
00:05the range, the lower quartile,
00:06the Q2, upper quartile, into
00:08quartile range, standard deviation, and
00:10the variance. And I'm going
00:12to do them using this
00:13example, the economics grades of
00:1510 IP students were recorded
00:17and given below. So these
00:19are the grades. So I'm
00:20going to four. So I'm
00:22a six, a two, a
00:23seven, a six, a three,
00:23a five, and a seven.
00:25Fine. I'm going to do,
00:27I'm going to
00:28calculate all of these things
00:29well as many as I
00:31can without a calculator to
00:33try and hopefully improve understanding
00:37and remember if they might
00:40make a situation where they
00:42can put in a grade
00:43like say one of the
00:44grades is k and they
00:46tell you the average and
00:47then you have to find
00:47it so it's important you
00:49know how to do it
00:50without a calculator. So A
00:52is the mean, how do
00:53I get the mean? Well
00:54firstly how do we write
00:56the mean. X bar, get
00:58used to seeing that, that's
00:59the way to write the
00:59mean. X bar, that's just
01:01X with a line on
01:01top of it. Or mu,
01:04the Greek letter mu is
01:05what we use for the
01:06mean. And what it is
01:07is we sum these up
01:09and divide by 10, because
01:10there's 10 students. So I'm
01:12going to do 4 plus
01:133 plus 3 plus 6
01:17plus 2, 7 plus 6
01:20plus 3 plus 5 plus
01:237.
01:24All over 10, which gives
01:28me 4 plus 3 plus
01:303 is 10, 16, 17,
01:3218, 25, 31, 34, 39,
01:3646. So it's 46 over
01:3910, which is 4 .6.
01:43Okay, great, B. Find the
01:46mode. What is the mode?
01:48The mode is the most
01:49common, the most common.
01:52the most common grade in
01:54this bunch of 10. Well,
01:55two people got a seven
01:57and three people got a
01:59three. So it's actually three.
02:03So the mode is mode
02:05equals three. Now it's three
02:08because the grade is three,
02:09not because there's three three's.
02:11Probably should have done a
02:12different number, but just be
02:13very clear. The mode is
02:15three. If there were four
02:16sevens, the mode would be
02:17seven.
02:20Let's see the median. Okay,
02:23to find the median, the
02:25median is the middle number.
02:27It's the, it's going to
02:29be the middle one, but
02:30when they're arranged in order.
02:32So first arrange them in
02:34order. So we have a
02:34two, that's the lowest, the
02:36two. And then we have
02:37a three, a three, a
02:40three, we have a four,
02:42we have one five, two
02:45sixes, two sixes.
02:48sixes and two sevens. Okay,
02:52who's the median? Who's in
02:54the middle? Well, there's one,
02:57two, three, four, five and
02:59one, two, three, four, five.
03:01So the median is actually
03:03right here. This is the
03:06median, the median. And it's
03:10halfway between four and five.
03:12And if it's halfway between
03:13four and five, the median
03:15then is
03:16median is 4 .5 because
03:21it's, you want to do
03:224 plus 5 over 2,
03:24which is 4 .5. It's
03:26halfway between 4 and 5.
03:28If there was a number
03:28here, like another 4 for
03:33example, then 4 would be
03:34the median. Okay, D, the
03:38range. Let's go down here.
03:41What is the range? So
03:42the range is the biggest
03:43minus the smallest.
03:44So simply 7 minus 2
03:47equals 5. The range is
03:495. Easy. E. The lower
03:53quartile. Okay. So the way
03:54we get the, the way
03:55we get the quartiles is
03:57if this is the median
03:58right in the middle, the
04:00quartile, the lower quartile, this
04:02is like halfway. This is
04:04like halfway through the data.
04:07The q1 is a quarter
04:08of the way through the
04:09data. So I need to
04:09find a quarter of the
04:11way through the data, which
04:12is actually
04:12like it's a bit like
04:14the median of these five
04:16to find the first quarter
04:17and it's actually going to
04:19be here. He's right in
04:21the middle because look there's
04:23two to the left and
04:24two to the right so
04:25this guy is Q1 and
04:27then similarly here what's the
04:29median of these five numbers?
04:32Well it's this one so
04:33this guy's Q3. Now median
04:36I've actually put Q2 here
04:38just as a trick question
04:40just to be clear that
04:42you understand the median is
04:44Q2. So you have Q1,
04:47Q2 is the median and
04:48Q3. So, e, the lower
04:52quartile, Q1 equals 3. F,
04:58Q2, that's Q1. Q2 equals
05:034 .5, it's the same
05:04as the median. And Q3
05:08equals 3.
05:08Three, Q3 is six. Okay,
05:15H, E, F, G, H.
05:18The interquartile range, I, Q,
05:21or is the range between
05:23the quartiles. It's Q3 minus
05:27Q1. Let's Q3 minus Q1,
05:31six minus three, which is
05:35three.
05:366 minus 3 is 3.
05:38That's the intercorder range. And
05:41then standard deviation and variance,
05:43okay, I can't get these
05:46without a calculator or at
05:47least you don't need to
05:48know how to do it.
05:49I actually have another video
05:51where I do it without
05:52a calculator just to kind
05:54of show you how it's
05:55done because I think it's
05:56quite, I think it is
05:58quite interesting and it kind
06:01of helps you understand what
06:02they are. The standard deviation
06:03and the variance
06:04Well, certainly for variants that
06:07as the name suggests is
06:08a way of deciding how
06:14spread out the data is.
06:16So if everyone got a,
06:18let's say everyone got a
06:19seven, the variants and standard
06:21deviation is actually zero, because
06:22there's no spread at all.
06:24There's no spread, the data
06:25is not spread out. It's
06:26all at seven. But here
06:28we're going to have, but
06:28we're definitely going to have
06:29some bit of a spread
06:30because some we got a
06:31two and some we got
06:31a seven.
06:33I'm just gonna find these
06:34with the calculator and while
06:35I'm at it I'm gonna
06:36show you how we how
06:37we can find all the
06:38rest of them with a
06:39calculator or well I'll at
06:41least show you what we
06:42can find so Home I'm
06:45gonna open this spreadsheet now.
06:48I'm gonna do well firstly,
06:50let's just name this X
06:53It's nice to name the
06:55column that we're gonna do
06:56so X whatever or you
06:57can even call it grades
07:01Now we're going to put
07:02in the numbers. You don't
07:04have to put them in
07:04order. You just put, well,
07:06I'll put them in this
07:07order, but it doesn't have
07:08to be from smallest to
07:09biggest. So it's 4, 3,
07:123, 6, 2, 7, 6,
07:183, 5, 7. Okay, fine.
07:24Now I'm going to do
07:25menu, statistics, and I'm doing
07:29a
07:29calculations and I'm doing one
07:31variable statistics because I have
07:32one variable and I want
07:33to see the statistics for
07:35this one variable. Press OK.
07:39The x list, I click
07:40this arrow and it's grades.
07:43It's the grades I want.
07:44There's no frequency list so
07:45just leave that. That's fine.
07:47And this is results in
07:48b, fine. It'll come in
07:49the second column. Press OK.
07:52OK. Now, what are these
07:55things? X bar, that's the
07:57mean 4 .6. That's what
07:59I got here. And you
08:00see X bar, X bar.
08:03This means the sum of
08:05X when you add the
08:05mall up, fine, but we
08:06don't need that. And the
08:07sum of X squared, I
08:08don't need this. S X,
08:11that's, we don't use that,
08:13right? That's a way of,
08:14that's another standard deviation that
08:16we're not using. This is
08:18the standard deviation. And this
08:22is the letter sigma. So
08:24it's like sigma X.
08:25So it's 1 .74356. So
08:29I'm going to write sigma
08:31equals 1 .74356. That is
08:39the standard deviation. And that's,
08:42as I said, a measure
08:43of how spread the data
08:45is. Until you watch my
08:46next lesson, it's hard to
08:47explain exactly what that means.
08:50But it means there's definitely
08:52some kind of spread.
08:53n is the number of
09:00students in this case. Minimum
09:02is 2, so it doesn't
09:07give me the range, but
09:08it gives me the min
09:08and the max, so I
09:09can use that easily to
09:10find the min and the
09:11max. But that's not that
09:13difficult to do anyway. Q1,
09:16that's my lower quarter, it's
09:183. Median is 4 .5,
09:20which I got.
09:21Q3 is 6 which I
09:23got. The max is 7
09:25which we know. This don't
09:27worry about it and that's
09:29it. So that's all the
09:30kind of statistics it gives
09:31you. So note it didn't
09:33give me the inter -quarter
09:34range but look you know
09:35that's Q3 minus Q1. It
09:37didn't give me the mode
09:38but you just have to
09:39look at which is the
09:40most common and it didn't
09:41give me the range but
09:42you can just do max
09:43minus min and that will
09:45give you the range. Okay
09:47that's it.
09:49I obviously did a lot
09:52there, but you can see
09:55with a calculator that's actually
09:58not that difficult at all.
10:00Actually, sorry, I'm not finished.
10:02I didn't do the variance.
10:03So the variance is sigma
10:07squared, and this is important.
10:10So it doesn't give me
10:11the variance here. No word
10:13is a good bit of
10:13variance, but the variance, it
10:15gives me the standard deviation.
10:17And that variance is just
10:19the standard deviation squared. So
10:21it's 1 .7IENCE 3 .56
10:25squared, which equals 1 .7IENCE
10:334 .56. I just want
10:34to make sure that was
10:35correct. Let's go to this.
10:39So 1 .7IENCE 4 .56.
10:45squared is 3 .04 exactly
10:49nice. So the variance is
10:523 .04. Okay, that's it
10:57done. So it doesn't give
10:59me the variance, but I
11:00can calculate it by squaring
11:01the standard deviation. Fine, it
11:04doesn't give me the integral
11:05range, but it gives me
11:07Q1 and Q3, and I
11:08just need to subtract them.
11:09And it doesn't give me
11:10the range or the mode,
11:11but I can find out
11:13those.
11:13Yes, I'd like you to
11:15know how to do certainly
11:17A to H without a
11:19calculator.